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List of NHL statistical leaders
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Contents
Skaters
All-time leaders
Regular season: Points
Regular season: Points per Game
Regular season: Goals
Regular season: Goals per Game
Regular season: Powerplay Goals
Regular season: Short-handed Goals
Regular season: Game-winning Goals
Regular season: Overtime Goals
Regular season: Assists
Regular season: Assists per game
Regular season: Games Played
Regular season: Penalty Minutes
Regular season: Plus-Minus
Regular season: Shots on Goal
Regular season: Shooting Percentage
Playoff points
Playoff points per game
Playoff goals
Playoff goals per game
Playoff powerplay goals
Playoff short-handed goals
Playoff game-winning goals
Playoff overtime goals
Playoff assists
Playoff games played
Playoff penalty minutes
Playoff plus-minus
Playoff shots on goal
Playoff shooting percentage
Active leaders
Regular season points
Regular season points per game
Regular season goals
Regular season goals per game
Regular season powerplay goals
Regular season short-handed goals
Regular season game-winning goals
Regular season overtime goals
Regular season assists
Regular season assists per game
Regular season games played
Regular season penalty minutes
Regular season plus-minus
Regular season shots on goal
Regular season shooting percentage
Playoff points
Playoff points per game
Playoff goals
Playoff goals per game
Playoff powerplay goals
Playoff short-handed goals
Playoff game-winning goals
Playoff overtime goals
Playoff assists
Playoff games played
Playoff penalty minutes
Playoff plus-minus
Playoff shots on goal
Playoff shooting percentage
Goaltenders
All-time leaders
Regular season wins
Regular season shutouts
Regular season goals against average
Regular season save percentage
Playoff wins
Playoff shutouts
Active leaders
Regular season wins
Regular season shutouts
Regular season goals against average
Regular season save percentage
Playoff wins
Playoff shutouts
Coaches
All-time leaders
Regular season games coached
Regular season coaching wins
Regular season coaching points percentage
Playoff games coached
Playoff coaching wins
Stanley Cups
Active leaders
Regular season games coached
Regular season coaching wins
Regular season coaching points percentage
Playoff games coached
Playoff coaching wins
Stanley Cups
External links
Skaters[edit]
The statistics listed include the 2015–16 NHL regular season and 2016 playoffs.
All-time leaders (skaters)[edit]
Active skaters (during 2015–16 NHL season) are listed in boldface.
Regular season: Points[edit]
Active NHL player
Inducted into the Hockey Hall of Fame
Eligible player not yet inducted into the Hockey Hall of Fame
Not yet eligible for Hockey Hall of Fame consideration[1]
Further information: Point (ice hockey)
Rank
Name
Team(s)
GP
Pts
PPG
1
Wayne Gretzky EDM, LAK, STL, NYR 1,487 2,857 1.92
2
Jaromir Jagr PIT, WSH, NYR, PHI, DAL, BOS, NJD, FLA 1,711 1,914 1.12
3
Mark Messier EDM, NYR, VAN 1,756 1,887 1.07
4
Gordie Howe DET, HFD 1,767 1,850 1.05
5
Ron Francis HFD, PIT, CAR, TOR 1,731 1,798 1.04
6
Marcel Dionne DET, LAK, NYR 1,348 1,771 1.31
7
Steve Yzerman DET 1,514 1,755 1.16
8
Mario Lemieux PIT 915 1,723 1.88
9
Joe Sakic QUE/COL 1,378 1,641 1.19
10
Phil Esposito CHI, BOS, NYR 1,282 1,590 1.24
11
Ray Bourque BOS, COL 1,612 1,579 0.98
12
Mark Recchi PIT, PHI, MTL, CAR, ATL, TBL, BOS 1,652 1,533 0.93
13
Paul Coffey EDM, PIT, LAK, DET, HFD, PHI, CHI, CAR, BOS 1,409 1,531 1.09
14
Stan Mikita CHI 1,394 1,467 1.05
15
Teemu Selanne WPG, ANA, SJS, COL 1,451 1,457 1.00
16
Bryan Trottier NYI, PIT 1,279 1,425 1.11
17
Adam Oates DET, STL, BOS, WSH, PHI, ANA, EDM 1,337 1,420 1.06
18
Doug Gilmour STL, CGY, TOR, NJD, CHI, BUF, MTL 1,474 1,414 0.96
19
Dale Hawerchuk WPG, BUF, STL, PHI 1,188 1,409 1.19
20
Jari Kurri EDM, LAK, NYR, ANA, COL 1,251 1,398 1.12
21
Luc Robitaille LAK, PIT, NYR, DET 1,431 1,394 0.97
22
Brett Hull CGY, STL, DAL, DET, PHX 1,269 1,391 1.10
23
Mike Modano MNS/DAL, DET 1,499 1,374 0.92
24
Johnny Bucyk DET, BOS 1,540 1,369 0.89
25
Brendan Shanahan NJD, STL, HFD, DET, NYR 1,524 1,354 0.89
26
Guy Lafleur MTL, NYR, QUE 1,127 1,353 1.20
27
Mats Sundin QUE, TOR, VAN 1,346 1,349 1.00
28
Joe Thornton BOS, SJ 1,367 1,341 0.98
29
Dave Andreychuk BUF, TOR, NJD, BOS, COL, TBL 1,639 1,338 0.82
30
Denis Savard CHI, MTL, TBL 1,196 1,338 1.12
31
Mike Gartner WSH, MNS, NYR, TOR, PHX 1,432 1,335 0.93
32
Pierre Turgeon BUF, NYI, MTL, STL, DAL, COL 1,294 1,327 1.03
33
Gilbert Perreault BUF 1,191 1,326 1.11
34
Jarome Iginla CGY, PIT, BOS, COL, LAK 1,554 1,300 0.86
35
Alex Delvecchio DET 1,549 1,281 0.83
36
Al MacInnis CGY, STL 1,416 1,274 0.90
37
Jean Ratelle NYR, BOS 1,281 1,267 0.99
38
Peter Stastny QUE, NJD, STL 977 1,239 1.27
39
Phil Housley BUF, WPG, STL, CGY, NJD, WSH, CHI, TOR 1,495 1,232 0.82
40
Norm Ullman DET, TOR 1,410 1,229 0.87
41
Jean Beliveau MTL 1,125 1,219 1.08
42
Larry Murphy LAK, WSH, MNS, PIT, TOR, DET 1,615 1,216 0.75
43
Jeremy Roenick CHI, PHX, PHI, LAK, SJS 1,363 1,216 0.89
44
Bobby Clarke PHI 1,144 1,210 1.06
45
Bernie Nicholls LAK, NYR, EDM, NJD, CHI, SJS 1,127 1,209 1.07
46
Vincent Damphousse TOR, EDM, MTL, SJS, COL 1,378 1,205 0.87
47
Dino Ciccarelli MNS, WSH, DET, TBL, FLA 1,232 1,200 0.97
48
Rod Brind'Amour STL, PHI, CAR 1,484 1,184 0.80
49
Sergei Fedorov DET, ANA, CBJ, WSH 1,248 1,179 0.94
50
Bobby Hull CHI, WPG, HFD 1,063 1,170 1.10
Regular season: Points per Game[edit]
Further information: Point (ice hockey)
Minimum 500 points
Wayne Gretzky, 1.921
Mario Lemieux, 1.883
Mike Bossy, 1.497
Bobby Orr, 1.393
Sidney Crosby, 1.327
Marcel Dionne, 1.314
Peter Stastny, 1.268
Peter Forsberg, 1.250
Kent Nilsson, 1.241
Phil Esposito, 1.240
Guy Lafleur, 1.202
Joe Sakic, 1.191
Dale Hawerchuk, 1.186
Evgeni Malkin, 1.180
Pat LaFontaine, 1.171
Steve Yzerman, 1.159
Alexander Ovechkin, 1.151
Jaromir Jagr, 1.147
Eric Lindros, 1.138
Bernie Federko, 1.130
Denis Savard, 1.119
Jari Kurri, 1.118
Bryan Trottier, 1.114
Gilbert Perreault, 1.113
Pavel Bure, 1.110
Regular season: Goals[edit]
Further information: Goal (ice hockey)
Wayne Gretzky, 894
Gordie Howe, 801
Jaromir Jagr, 765
Brett Hull, 741
Marcel Dionne, 731
Phil Esposito, 717
Mike Gartner, 708
Mark Messier, 694
Steve Yzerman, 692
Mario Lemieux, 690
Teemu Selanne, 684
Luc Robitaille, 668
Brendan Shanahan, 656
Dave Andreychuk, 640
Joe Sakic, 625
Jarome Iginla, 625
Bobby Hull, 610
Dino Ciccarelli, 608
Jari Kurri, 601
Mark Recchi, 577
Mike Bossy, 573
Joe Nieuwendyk, 564
Mats Sundin, 564
Mike Modano, 561
Guy Lafleur, 560
Alexander Ovechkin, 558
Johnny Bucyk, 556
Ron Francis, 549
Michel Goulet, 548
Maurice Richard, 544
Stan Mikita, 541
Keith Tkachuk, 538
Frank Mahovlich, 533
Marian Hossa, 525
Bryan Trottier, 524
Pat Verbeek,522
Dale Hawerchuk, 518
Pierre Turgeon, 515
Jeremy Roenick, 513
Gilbert Perreault, 512
Jean Beliveau, 507
Peter Bondra, 503
Joe Mullen, 502
Lanny McDonald, 500
Glenn Anderson, 498
Jean Ratelle, 491
Norm Ullman, 490
Brian Bellows, 485
Darryl Sittler, 484
Sergei Fedorov, 483
Regular season: Goals per Game[edit]
Minimum: 200 goals
Mike Bossy, 0.762
Cy Denneny, 0.756
Mario Lemieux, 0.754
Babe Dye, 0.742
Alexander Ovechkin, 0.626
Pavel Bure, 0.623
Wayne Gretzky, 0.601
Brett Hull, 0.584
Bobby Hull, 0.574
Tim Kerr, 0.565
Rick Martin, 0.561
Phil Esposito, 0.559
Maurice Richard, 0.556
Steven Stamkos, 0.548
Cam Neely, 0.544
Marcel Dionne, 0.542
Pat LaFontaine, 0.541
Ilya Kovalchuk, 0.511
Rick Vaive, 0.503
Michel Goulet, 0.503
Nels Stewart, 0.498
Guy Lafleur, 0.497
Mike Gartner, 0.494
Dino Ciccarelli, 0.493
Howie Morenz, 0.493
Regular season: Power Play Goals[edit]
Further information: Power play (sporting term)
Dave Andreychuk, 274
Brett Hull, 265
Teemu Selanne, 255
Phil Esposito, 249
Luc Robitaille, 247
Brendan Shanahan, 237
Mario Lemieux, 236
Marcel Dionne, 234
Dino Ciccarelli, 232
Mike Gartner, 217
Joe Nieuwendyk, 215
Keith Tkachuk, 212
Jaromir Jagr, 209
Joe Sakic, 205
Wayne Gretzky, 204
Steve Yzerman, 202
Mark Recchi, 200
Brian Bellows, 198
Alexander Ovechkin, 195
Jarome Iginla, 197
Pierre Turgeon, 190
Ron Francis, 188
Pat Verbeek, 186
Jeremy Roenick, 184
Dale Hawerchuk, 182
Regular season: Short-handed goals[edit]
Further information: Short-handed
Wayne Gretzky, 73
Mark Messier, 63
Steve Yzerman, 50
Mario Lemieux, 49
Butch Goring, 40
Dave Poulin, 39
Jari Kurri, 37
Sergei Fedorov, 36
Theoren Fleury, 35
Dirk Graham, 35
Pavel Bure, 34
Derek Sanderson, 34
Marian Hossa, 33
Brian Rolston, 33
Peter Bondra, 32
Guy Carbonneau, 32
Bobby Clarke, 32
Joe Sakic, 32
Bill Barber, 31
Mats Sundin, 31
Martin St. Louis, 29
Russ Courtnall, 29
Craig MacTavish, 29
Mike Modano, 29
Esa Tikkanen, 29
Regular season: Game-winning goals[edit]
Jaromir Jagr, 133
Phil Esposito, 118
Brett Hull, 110
Teemu Selanne, 110
Brendan Shanahan, 109
Jarome Iginla, 101
Guy Lafleur, 97
Mats Sundin, 96
Steve Yzerman, 94
Sergei Fedorov, 93
Joe Nieuwendyk, 93
Mark Messier, 92
Mike Modano, 92
Jeremy Roenick, 92
Patrick Marleau, 92
Wayne Gretzky, 91
Mark Recchi, 91
Mike Gartner, 90
Luc Robitaille, 89
Alexander Ovechkin, 88
Joe Sakic, 86
Pierre Turgeon, 86
Glenn Anderson, 85
Mike Bossy, 82
Gilbert Perreault, 81
Regular season: Overtime goals[edit]
If a game is tied after regulation time (which lasts three 20-minutes periods), there will be a period of "overtime" to decide the winner. The player who scores during this extra time is given the overtime goal. All overtime in the NHL is sudden death—meaning the first team to score is the winner—so the player who scores in overtime also has the game-winning goal.
Jaromir Jagr, 19
Alexander Ovechkin, 17
Patrik Elias, 16
Mats Sundin, 15
Sergei Fedorov, 15
Ilya Kovalchuk, 14
Daniel Sedin, 14
Scott Niedermayer, 13
Olli Jokinen, 13
Marian Hossa, 12
Daniel Briere, 12
Brendan Shanahan, 12
Brett Hull, 12
Shane Doan, 11
Marian Gaborik, 11
Theoren Fleury, 11
Pierre Turgeon, 11
Glen Murray, 11
Brendan Morrison, 10
Saku Koivu, 10
Steve Thomas, 10
Martin St. Louis, 10
Evgeni Malkin, 9
Mike Modano, 9
Sidney Crosby, 9
Milan Hejduk, 9
Loui Eriksson, 9
Teemu Selanne, 9
Pavol Demitra, 9
Vincent Lecavalier, 9
Bill Guerin, 9
David Legwand, 9
Joe Thornton, 9
Patrick Marleau, 9
Regular season: Assists[edit]
Further information: Assist (ice hockey)
Wayne Gretzky, 1,963
Ron Francis, 1,249
Mark Messier, 1,193
Ray Bourque, 1,169
Jaromir Jagr, 1,149
Paul Coffey, 1,135
Adam Oates, 1,079
Steve Yzerman, 1,063
Gordie Howe, 1,049
Marcel Dionne, 1,040
Mario Lemieux, 1,033
Joe Sakic, 1,016
Joe Thornton, 1,007
Doug Gilmour, 964
Mark Recchi, 956
Al MacInnis, 934
Larry Murphy, 929
Stan Mikita, 926
Bryan Trottier, 901
Phil Housley, 894
Dale Hawerchuk, 891
Nicklas Lidstrom, 878
Phil Esposito, 873
Denis Savard, 865
Bobby Clarke, 852
Alex Delvecchio, 825
Gilbert Perreault, 814
Johnny Bucyk, 813
Mike Modano, 813
Pierre Turgeon, 812
Jari Kurri, 797
Guy Lafleur, 793
Peter Stastny, 789
Henrik Sedin, 783
Mats Sundin, 785
Brian Leetch, 781
Jean Ratelle, 776
Vincent Damphousse, 773
Teemu Selanne, 773
Chris Chelios, 763
Bernie Federko, 761
Doug Weight, 755
Larry Robinson, 750
Denis Potvin, 742
Norm Ullman, 739
Bernie Nicholls, 734
Rod Brind'Amour, 732
Luc Robitaille, 726
Daniel Alfredsson, 713
Jean Beliveau, 712
Scott Stevens, 712
Regular season: Assists per game[edit]
Minimum: 300 assists
Wayne Gretzky, 1.320
Mario Lemieux, 1.129
Bobby Orr, 0.982
Peter Forsberg, 0.898
Sidney Crosby, 0.849
Peter Stastny, 0.808
Adam Oates, 0.807
Paul Coffey, 0.806
Marcel Dionne, 0.772
Kent Nilsson, 0.763
Bernie Federko, 0.761
Dale Hawerchuk, 0.750
Bobby Clarke, 0.745
Craig Janney, 0.741
Joe Sakic, 0.737
Mike Bossy, 0.735
Nicklas Backstrom, 0.732
Ray Bourque, 0.725
Denis Savard, 0.723
Evgeni Malkin, 0.7220
Ron Francis, 0.7215
Joe Thornton, 0.705
Bryan Trottier, 0.7045
Guy Lafleur, 0.7043
Steve Yzerman, 0.702
Regular season: Games played[edit]
Further information: Games played
Gordie Howe, 1,767
Mark Messier, 1,756
Ron Francis, 1,731
Jaromir Jagr, 1,711
Mark Recchi, 1,652
Chris Chelios, 1,651
Dave Andreychuk, 1,639
Scott Stevens, 1,635
Larry Murphy, 1,615
Ray Bourque, 1,612
Nicklas Lidstrom, 1,564
Jarome Iginla, 1,554
Alex Delvecchio, 1,549
Johnny Bucyk, 1,540
Brendan Shanahan, 1,524
Steve Yzerman, 1,514
Mike Modano, 1,499
Phil Housley, 1,495
Wayne Gretzky, 1,487
Rod Brind'Amour, 1,484
Doug Gilmour, 1,474
Shane Doan, 1,456
Glen Wesley, 1,457
Teemu Selanne, 1,451
Tim Horton, 1,446
Regular season: Penalty minutes[edit]
Further information: Penalty (ice hockey)
Tiger Williams, 3,966
Dale Hunter, 3,565
Tie Domi, 3,515
Marty McSorley, 3,381
Bob Probert, 3,300
Robert Ray, 3,207
Craig Berube, 3,149
Tim Hunter, 3,146
Chris Nilan, 3,043
Rick Tocchet, 2,972
Pat Verbeek, 2,905
Chris Chelios, 2,891
Dave Manson, 2,792
Scott Stevens, 2,785
Donald Brashear, 2,634
Willi Plett, 2,572
Gino Odjick, 2,567
Matthew Barnaby, 2,562
Gary Roberts, 2,560
Joey Kocur, 2,519
Kenneth Daneyko, 2,516
Brendan Shanahan, 2,489
Scott Mellanby, 2,479
Chris Neil, 2,459
Basil McRae, 2,457
Regular season: Plus-minus[edit]
Further information: Plus-minus
Larry Robinson, +730
Bobby Orr, +597
Ray Bourque, +528
Wayne Gretzky, +518
Bobby Clarke, +506
Denis Potvin, +460
Serge Savard, +460
Guy Lafleur, +453
Bryan Trottier, +452
Nicklas Lidstrom, +450
Brad McCrimmon, +444
Mark Howe, +400
Steve Shutt, +393
Mike Bossy, +381
Scott Stevens, +377
Al MacInnis, +373
Brad Park, +358
Dallas Smith, +355
Chris Chelios, +350
Jacques Lemaire, +349
Guy Lapointe, +329
Craig Ramsay, +328
Bill Hajt, +321
Jaromir Jagr, +316
Bill Barber, +315
Regular season: Shots on goal[edit]
Further information: Shot on goal
Ray Bourque, 6,206
Jaromir Jagr, 5,424
Marcel Dionne, 5,366
Al MacInnis, 5,157
Mike Gartner, 5,090
Wayne Gretzky, 5,089
Brendan Shanahan, 5,086
Jarome Iginla, 4,756
Brett Hull, 4,876
Joe Sakic, 4,621
Steve Yzerman, 4,602
Phil Esposito, 4,595
Dave Andreychuk, 4,556
Teemu Selanne, 4,540
Paul Coffey, 4,385
Mike Modano, 4,273
Alexander Ovechkin, 4,228
Mark Messier, 4,219
Marian Hossa, 4,062
Mats Sundin, 4,015
Sergei Fedorov, 3,985
Luc Robitaille, 3,961
Rob Blake, 3,896
Nicklas Lidstrom, 3,875
Brian Rolston, 3,836
Regular season: Shooting percentage[edit]
Shooting percentage is the percentage of shots on goal which result in a goal.
Minimum 800 shots
Craig Simpson, 23.66%
Charlie Simmer, 22.34%
Paul MacLean, 21.41%
Mike Bossy, 21.18%
Yvon Lambert, 19.85%
Rick Middleton, 19.69%
Blaine Stoughton, 19.52%
Darryl Sutter, 19.42%
Rob Brown, 19.41%
Mike Ridley, 19.30%
Steve Vickers, 19.28%
Kent Nilsson, 19.21%
Tom McCarthy, 19.16%
Jari Kurri, 19.13%
Johnny Bucyk, 19.09%
Mario Lemieux, 18.99%
Peter Stastny, 18.96%
Ray Ferraro, 18.85%
Mark Hunter, 18.78%
Tim Kerr, 18.77%
Playoff: Points[edit]
Further information: Point (ice hockey)
Wayne Gretzky, 382
Mark Messier, 295
Jari Kurri, 233
Glenn Anderson, 214
Jaromir Jagr, 201
Paul Coffey, 196
Brett Hull, 190
Doug Gilmour, 188
Joe Sakic, 188
Steve Yzerman, 185
Bryan Trottier, 184
Nicklas Lidstrom, 183
Ray Bourque, 180
Jean Beliveau, 176
Sergei Fedorov, 176
Denis Savard, 175
Mario Lemieux, 172
Peter Forsberg, 171
Denis Potvin, 164
Mike Bossy, 160
Gordie Howe, 160
Al MacInnis, 160
Bobby Smith, 160
Claude Lemieux, 157
Adam Oates, 156
Playoff points per game[edit]
Minimum: 50 points
Wayne Gretzky, 1.837
Mario Lemieux, 1.607
Barry Pederson, 1.529
Mark Messier, 1.250
Bobby Orr, 1.243
Mike Bossy, 1.240
Jari Kurri, 1.180
Gilbert Perreault, 1.165
Peter Forsberg, 1.144
Peter Stastny, 1.132
Bernie Federko, 1.129
Sidney Crosby, 1.105
Pavel Bure, 1.094
Joe Sakic, 1.093
Jean Beliveau, 1.086
Bobby Hull, 1.084
Eric Lindros, 1.075
Toe Blake, 1.069
Ken Linseman, 1.062
Phil Esposito, 1.054
Guy Lafleur, 1.047
Evgeni Malkin, 1.040
Denis Savard, 1.036
Doug Gilmour, 1.033
Kevin Stevens, 1.029
Playoff goals[edit]
Further information: Goal (ice hockey)
Wayne Gretzky, 122
Mark Messier, 109
Jari Kurri, 106
Brett Hull, 103
Glenn Anderson, 93
Mike Bossy, 85
Joe Sakic, 84
Maurice Richard, 82
Claude Lemieux, 80
Jean Beliveau, 79
Jaromir Jagr, 78
Mario Lemieux, 76
Dino Ciccarelli, 73
Esa Tikkanen, 72
Bryan Trottier, 71
Steve Yzerman, 70
Gordie Howe, 68
Patrick Marleau, 68
Joe Nieuwendyk, 66
Denis Savard, 66
Yvan Cournoyer, 64
Peter Forsberg, 64
Brian Propp, 64
Bobby Smith, 64
Bobby Hull, 62
Playoff goals per game[edit]
Minimum 20 goals
Mario Lemieux, 0.710
Mike Bossy, 0.659
Barry Pederson, 0.647
Maurice Richard, 0.617
Cam Neely, 0.613
Wayne Gretzky, 0.587
Pavel Bure, 0.547
Craig Simpson, 0.537
Jari Kurri, 0.530
Bobby Hull, 0.521
Gordie Drillon, 0.520
Dino Ciccarelli, 0.518
Brett Hull, 0.510
Steve Shutt, 0.505
Reggie Leach, 0.500
Rick Vaive, 0.500
Phil Kessel, 0.500
Tim Kerr, 0.494
Steve Payne, 0.493
Nikita Kucherov, 0.489
Playoff Power Play goals[edit]
Further information: Power play (sporting term)
When a team is given a penalty for committing an infraction (such as tripping another player), the offending player must sit in the penalty box, and his team must play with one fewer player on the ice. The penalized team is said to be "short-handed", while the other team has a "powerplay". If a player scores while his team is on the powerplay, this is recorded as a powerplay goal.
Brett Hull, 38
Mike Bossy, 35
Dino Ciccarelli, 34
Wayne Gretzky, 34
Nicklas Lidstrom, 30
Mario Lemieux, 29
Denis Potvin, 28
Brian Propp, 27
Joe Sakic, 27
Steve Yzerman, 27
Mark Messier, 26
Jari Kurri, 25
Cam Neely, 25
Daniel Alfredsson, 25
Jaromir Jagr, 24
Al MacInnis, 24
Mike Modano, 24
Denis Savard, 24
Bobby Smith, 24
Playoff short-handed goals[edit]
When a team is given a penalty for committing an infraction (such as tripping another player), the offending player must sit in the penalty box, and his team must play with one fewer player on the ice. The penalized team is said to be "short-handed", while the other team has a "powerplay". If a player scores while his team is short handed, this is recorded as a short-handed goal.
Mark Messier, 12
Wayne Gretzky, 11
Jari Kurri, 10
Hakan Loob, 8
Ed Westfall, 8
Mario Lemieux, 7
Paul Coffey, 7
Dave Poulin, 6
Wayne Presley, 6
Brian Rolston, 6
Derek Sanderson, 6
Bill Barber, 5
Bob Bourne, 5
Sergei Fedorov, 5
Lorne Henning, 5
Anders Kallur, 5
Kirk Maltby, 5
Kelly Miller,
Guy Carbonneau, 5
Playoff game-winning goals[edit]
Wayne Gretzky, 24
Brett Hull, 24
Claude Lemieux, 19
Joe Sakic, 19
Glenn Anderson, 17
Mike Bossy, 17
Chris Drury, 17
Jaromir Jagr, 16
Patrick Marleau, 16
Mike Modano, 15
Yvan Cournoyer, 15
Peter Forsberg, 14
Jari Kurri, 14
Guy Lafleur, 14
Joe Nieuwendyk, 14
Mark Messier, 14
Daniel Briere, 13
Dino Ciccarelli, 13
Doug Gilmour, 13
Stephane Richer, 13
Bobby Smith, 13
Playoff overtime goals[edit]
If a game is tied after regulation time (which lasts three 20-minutes periods), there will be as many 20 minute periods of "overtime" as necessary during the playoffs to determine a winner. The player who scores during this extra time is given the overtime goal. All overtime in the NHL is sudden death—meaning the first team to score is the winner—so the player who scores in overtime also has the game-winning goal.
Joe Sakic, 8
Maurice Richard, 6
Glenn Anderson, 5
Patrick Kane, 5
Wayne Gretzky, 4
Chris Drury, 4
Dale Hunter, 4
Jaromir Jagr, 4
Jamie Langenbrunner, 4
Kirk Muller, 4
Joe Murphy, 4
Bob Nystrom, 4
Stephane Richer, 4
Jeremy Roenick, 4
Esa Tikkanen, 4
Greg Adams, 3
Patrice Bergeron, 3
Daniel Briere, 3
Alexandre Burrows, 3
Mike Bossy, 3
Dino Ciccarelli, 3
Bobby Clarke, 3
Peter Forsberg, 3
Danny Gare, 3
Mike Gartner, 3
Martin Gelinas, 3
Doug Gilmour, 3
Mel Hill, 3
Brett Hull, 3
Jacques Lemaire, 3
Claude Lemieux, 3
Lanny McDonald, 3
Brenden Morrow, 3
Ken Morrow, 3
Joe Pavelski, 3
Steve Payne, 3
Rene Robert, 3
Devin Setoguchi, 3
Cory Stillman, 3
Martin St. Louis, 3
Petr Sykora, 3
Niclas Wallin, 3
Scott Young, 3
Playoff assists[edit]
Further information: Assist (ice hockey)
Wayne Gretzky, 260
Mark Messier, 186
Ray Bourque, 139
Paul Coffey, 137
Nicklas Lidstrom, 129
Doug Gilmour, 128
Jari Kurri, 127
Sergei Fedorov, 124
Jaromir Jagr, 123
Glenn Anderson, 121,
Al MacInnis, 121
Larry Robinson, 116
Larry Murphy, 115
Steve Yzerman, 115
Adam Oates, 114
Chris Chelios, 113
Bryan Trottier, 113
Denis Savard, 109
Denis Potvin, 108
Peter Forsberg, 107
Joe Sakic, 104
Jean Beliveau, 97
Ron Francis, 97
Mario Lemieux, 96
Bobby Smith, 96
Playoff games played[edit]
Further information: Games played
Chris Chelios, 266
Nicklas Lidstrom, 263
Mark Messier, 236
Claude Lemieux, 234
Scott Stevens, 233
Guy Carbonneau, 231
Larry Robinson, 227
Glenn Anderson, 225
Kris Draper, 222
Bryan Trottier, 221
Mike Keane, 220
Larry Murphy, 215
Ray Bourque, 214
Kevin Lowe, 214
Wayne Gretzky, 208
Brett Hull, 202
Jaromir Jagr, 202
Scott Niedermayer, 202
Jari Kurri, 200
Steve Yzerman, 196
Playoff penalty minutes[edit]
A penalty is given to a player for committing an infraction during the game. The length of the penalty varies depending on the severity of the offence. The amount of penalty minutes recorded for statistical purposes are:
minor – 2 minutes
double minor – 4 minutes
major – 5 minutes
misconduct – 10 minutes
game misconduct – 10 minutes
Dale Hunter, 729
Chris Nilan, 541
Claude Lemieux, 529
Rick Tocchet, 471
Willi Plett, 466
Tiger Williams, 455
Glenn Anderson, 442
Tim Hunter, 426
Chris Chelios, 423
Dave Schultz, 412
Duane Sutter, 405
Scott Stevens, 402
Jim Peplinski, 382
Al Secord, 382
Marty McSorley, 374
Andre Dupont, 352
Basil McRae, 349
Dave Manson, 343
Terry O'Reilly, 335
Gary Roberts, 332
Playoff plus-minus[edit]
Plus-minus is a statistic that indicates the relative goal differential when a player is on the ice. If the player is on the ice when his team scores even-strength or short-handed, he is given +1; if he is on the ice when the opposing team scores even-strength or short-handed, he is given -1.
Charlie Huddy, +82
Jari Kurri, +73
Randy Gregg, +71
Wayne Gretzky, +67
Glenn Anderson, +63
Nicklas Lidstrom, +61
Paul Coffey, +56
Peter Forsberg, +54
Mark Howe, +54
Steve Smith, +49
Chris Chelios, +48
Scott Stevens, +48
Claude Lemieux, +42
Mark Messier, +42
Brian Rafalski, +42
Henrik Zetterberg, +41
Chris Pronger, +40
Sergei Fedorov, +38
Kevin Lowe, +38
Milan Lucic, +37
Playoff shots on goal[edit]
Further information: Shot on goal
Brett Hull, 803
Claude Lemieux, 730
Al MacInnis, 664
Nicklas Lidstrom, 656
Jaromir Jagr, 649
Ray Bourque, 648
Brendan Shanahan, 622
Sergei Fedorov, 603
Chris Chelios, 587
Steve Yzerman, 585
Joe Sakic, 582
Mark Messier, 569
Wayne Gretzky, 549
Paul Coffey, 536
Mike Modano, 507
Scott Stevens, 473
Jari Kurri, 469
Larry Murphy, 468
Glenn Anderson, 451
Marian Hossa, 450
Playoff shooting percentage[edit]
Shooting percentage is the percentage of shots on goal which result in a goal.
Minimum 80 shots
Craig Simpson, 33.64%
Ken Linseman, 23.21%
Andrew Brunette, 21.25%
Tim Kerr, 20.33%
Bernie Federko, 20.20%
Martin St. Louis, 20.00%
Cam Neely, 19.59%
Jari Kurri, 19.40%
Ray Ferraro, 19.27%
Paul MacLean, 19.10%
Mario Lemieux, 18.91%
Kevin Dineen, 18.85%
Claude Giroux, 18.58%
Peter Stastny, 18.58%
Thomas Vanek, 18.29%
Peter Forsberg, 18.13%
Rick Vaive, 17.83%
Pat LaFontaine, 17.81%
Michel Goulet, 17.72%
John Druce, 17.53%
Active leaders (skaters)[edit]
Regular season points (active)[edit]
Further information: Point (ice hockey)
Jaromir Jagr, 1,914
Joe Thornton, 1,391
Jarome Iginla, 1,300
Marian Hossa, 1,134
Patrick Marleau, 1,082
Alexander Ovechkin, 1,035
Sidney Crosby, 1,027
Henrik Sedin, 1,020
Daniel Sedin, 986
Shane Doan, 972
Henrik Zetterberg, 904
Jason Spezza, 862
Eric Staal, 846
Evgeni Malkin, 832
Ryan Getzlaf, 814
Regular season points per game (active)[edit]
Minimum 500 points
Sidney Crosby, 1.327
Evgeni Malkin, 1.180
Alexander Ovechkin, 1.151
Jaromir Jagr, 1.147
Patrick Kane, 1.008
Steven Stamkos, 0.987
Nicklas Backstrom, 0.985
Joe Thornton, 0.980
Jason Spezza, 0.975
Ryan Getzlaf, 0.955
Henrik Zetterberg, 0.940
Marian Hossa, 0.901
Jonathan Toews, 0.896
Anze Kopitar, 0.893
Jarome Iginla, 0.881
Regular season goals (active)[edit]
Further information: Goal (ice hockey)
Jaromir Jagr, 765
Jarome Iginla, 625
Alexander Ovechkin, 558
Marian Hossa, 525
Patrick Marleau, 508
Rick Nash, 416
Shane Doan, 402
Marian Gaborik, 396
Joe Thornton, 384
Sidney Crosby, 382
Daniel Sedin, 370
Eric Staal, 353
Corey Perry, 349
Jeff Carter, 339
Thomas Vanek, 333
Evgeni Malkin, 328
Henrik Zetterberg, 326
Steven Stamkos, 321
Zach Parise, 318
Jason Spezza, 316
Regular season goals per game (active)[edit]
Minimum 200 goals
Alexander Ovechkin, 0.625
Steven Stamkos, 0.561
Sidney Crosby, 0.482
Jaromir Jagr, 0.466
Evgeni Malkin, 0.457
Rick Nash, 0.439
Dany Heatley, 0.428
Marian Gaborik, 0.425
Jarome Iginla, 0.423
Marian Hossa, 0.415
Corey Perry, 0.410
Thomas Vanek, 0.401
Zach Parise, 0.397
Jonathan Toews, 0.395
Jeff Carter, 0.394
Alexander Semin, 0.375
Phil Kessel, 0.370
Eric Staal, 0.369
Max Pacioretty, 0.361
Mike Cammalleri, 0.357
Regular season Power Play goals (active)[edit]
When a team is given a penalty for committing an infraction (such as tripping another player), the offending player must sit in the penalty box, and his team must play with one fewer player on the ice. The penalized team is said to be "short-handed", while the other team has a "powerplay". If a player scores while his team is on the powerplay, this is recorded as a powerplay goal.
Jaromir Jagr, 209
Alexander Ovechkin, 195
Jarome Iginla, 190
Patrick Marleau, 153
Marian Hossa, 138
Joe Thornton, 129
Shane Doan, 125
Thomas Vanek, 124
Daniel Sedin, 112
Regular season short-handed goals (active)[edit]
When a team is given a penalty for committing an infraction (such as tripping another player), the offending player must sit in the penalty box, and his team must play with one fewer player on the ice. The penalized team is said to be "short-handed", while the other team has a "powerplay". If a player scores while his team is short handed, this is recorded as a short-handed goal.
Marian Hossa, 33
Antoine Vermette, 22
Rick Nash, 21
Brad Marchand, 19
Tomas Plekanec, 18
Alexandre Burrows, 17
Patrick Marleau, 17
Matt Cullen, 17
Regular season game-winning goals (active)[edit]
Jaromir Jagr, 133
Jarome Iginla, 97
Patrick Marleau, 92
Alexander Ovechkin, 88
Daniel Sedin, 78
Marian Hossa, 78
Shane Doan, 69
Marian Gaborik, 68
Rick Nash, 68
Jeff Carter, 64
Henrik Zetterberg, 62
Joe Thornton, 61
Regular season overtime goals (active)[edit]
If a game is tied after regulation time (which lasts three 20-minutes periods), there will be a period of "overtime" to decide the winner. The player who scores during this extra time is given the overtime goal. All overtime in the NHL is sudden death—meaning the first team to score is the winner—so the player who scores in overtime also has the game-winning goal.
Jaromir Jagr, 19
Alexander Ovechkin, 17
Daniel Sedin, 14
Marian Hossa, 12
Shane Doan, 11
Marian Gaborik, 11
Loui Eriksson, 9
Sidney Crosby, 9
Evgeni Malkin, 9
Joe Thornton, 9
Patrick Marleau, 9
Regular season assists (active)[edit]
Further information: Assist (ice hockey)
Jaromir Jagr, 1,149
Joe Thornton, 1,007
Henrik Sedin, 783
Jarome Iginla, 675
Sidney Crosby, 645
Daniel Sedin, 616
Marian Hossa, 609
Henrik Zetterberg, Ryan Getzlaf, 578
Patrick Marleau, 574
Shane Doan, 570
Mike Ribeiro, 565
Jason Spezza, 546
Nicklas Backstrom, 540
Evgeni Malkin, 504
Eric Staal, 493
Anze Kopitar, 481
Alex Ovechkin, 477
Regular season assists per game (active)[edit]
Minimum 300 assists
Sidney Crosby, 0.879
Nicklas Backstrom, 0.740
Evgeni Malkin, 0.739
Joe Thornton, 0.701
Jaromir Jagr, 0.697
Ryan Getzlaf, 0.662
Henrik Sedin, 0.645
Pavel Datsyuk, 0.644
Jason Spezza, 0.626
Claude Giroux, 0.617
Patrick Kane, 0.611
Henrik Zetterberg, 0.586
Brad Richards, 0.582
Anze Kopitar, 0.574
Alexander Ovechkin, 0.553
Alex Tanguay, 0.543
Scott Gomez, 0.543
Paul Stastny, 0.536
Mike Ribeiro, 0.529
Mikko Koivu, 0.523
Daniel Sedin, 0.522
Eric Staal, 0.508
Patrik Elias, 0.499
Ales Hemsky, 0.493
Marian Hossa, 0.486
Regular season games played (active)[edit]
Further information: Games played
Jaromir Jagr, 1,711
Jarome Iginla, 1,554
Shane Doan, 1,540
Patrick Marleau, 1,493
Joe Thornton, 1,446
Matt Cullen, 1,366
Zdeno Chara, 1,350
Marian Hossa, 1,309
Henrik Sedin, 1, 248
Daniel Sedin, 1,225
Regular season penalty minutes (active)[edit]
A penalty is given to a player for committing an infraction during the game. The length of the penalty varies depending on the severity of the offence. The amount of penalty minutes recorded for statistical purposes are:
minor – 2 minutes
double minor – 4 minutes
major – 5 minutes
misconduct – 10 minutes
game misconduct – 10 minutes
Chris Neil, 2,459
Zdeno Chara, 1,720
Scott Hartnell, 1,664
Chris Pronger, 1,590
Steve Ott, 1,475
Eric Boulton, 1,421
Cody McLeod 1,307
Shane Doan, 1,305
Derek Dorsett, 1,207
Jared Boll, 1,195
Dion Phaneuf, 1,143
Joe Thornton, 1,105
Jaromir Jagr, 1,101
Shawn Thornton, 1,036
Alexandre Burrows, 1,013
Regular season plus-minus (active)[edit]
Plus-minus is a statistic that indicates the relative goal differential when a player is on the ice. If the player is on the ice when his team scores even-strength or short-handed, he is given +1; if he is on the ice when the opposing team scores even-strength or they score a goal while he is on the powerplay, he is given -1.
Jaromir Jagr, +314
Marian Hossa, +238
Henrik Sedin, +214
Zdeno Chara, +200
Joe Thornton, +199
Jonathan Toews, +187
Daniel Sedin, +184
Chris Pronger, +183
Chris Kunitz, +183
Duncan Keith, +150
Sidney Crosby, +148
Henrik Zetterberg, +144
Ryan Getzlaf, +129
Brad Marchand, +138
Marc-Edouard Vlasic, +125
Regular season shots on goal (active)[edit]
Further information: Shot on goal
Jaromir Jagr, 5,424
Jarome Iginla, 4,606
Alexander Ovechkin, 4,228
Marian Hossa, 4,062
Shane Doan, 3,801
Patrick Marleau, 3,608
Rick Nash, 3,199
Henrik Zetterberg, 3,080
Eric Staal, 3,073
Daniel Sedin, 3,069
Regular season shooting percentage (active)[edit]
Shooting percentage is the percentage of shots on goal which result in a goal.
Minimum 800 shots
Steven Stamkos, 17.2%
Jiri Hudler, 15.1%
Jonathan Toews, 15.1%
Brad Marchand, 15.0%
Milan Lucic, 14.8%
Playoff points (active)[edit]
Further information: Point (ice hockey)
Jaromir Jagr, 201
Sidney Crosby, 164
Evgeni Malkin, 157
Marian Hossa, 149
Patrick Kane, 123
Joe Thornton, 123
Patrick Marleau, 120
Henrik Zetterberg, 120
Ryan Getzlaf, 118
Jonathan Toews, 110
Justin Williams, 94
Playoff points per game (active)[edit]
Minimum 50 points
Sidney Crosby, 1.180
Evgeni Malkin, 1.099
Claude Giroux, 1.070
Jaromir Jagr, 0.985
Patrick Kane, 0.983
Alexander Ovechkin, 0.972
Ryan Getzlaf, 0.969
Daniel Briere, 0.935
Jason Spezza, 0.929
Henrik Zetterberg, 0.901
Playoff goals (active)[edit]
Further information: Goal (ice hockey)
Jaromir Jagr, 78
Patrick Marleau, 68
Evgeni Malkin, 58
Henrik Zetterberg, 57
Sidney Crosby, 57
Marian Hossa, 52
Patrick Kane, 50
Patrick Sharp, 47
Alex Ovechkin, 46
Joe Pavelski, 42
Playoff goals per game (active)[edit]
Minimum 20 goals
Alexander Ovechkin, 0.500
Jarome Iginla, 0.457
Sidney Crosby, 0.430
Daniel Briere, 0.427
Henrik Zetterberg, 0.424
Evgeni Malkin, 0.416
Patrick Kane, 0.414
Patrick Marleau, 0.408
Claude Giroux, 0.404
Marian Gaborik, 0.400
Johan Franzen, 0.393
Jaromir Jagr, 0.386
Vincent Lecavalier, 0.357
Corey Perry, 0.356
Simon Gagne, 0.339
Playoff power play goals (active)[edit]
When a team is given a penalty for committing an infraction (such as tripping another player), the offending player must sit in the penalty box, and his team must play with one fewer player on the ice. The penalized team is said to be "short-handed," while the other team has a "powerplay." If a player scores while his team is on the powerplay, this is recorded as a powerplay goal.
Jaromir Jagr, 24
Patrick Marleau, 22
Evgeni Malkin, 22
Henrik Zetterberg, 21
Marian Hossa, 18
Joe Pavelski, 17
Playoff short-handed goals (active)[edit]
When a team is given a penalty for committing an infraction (such as tripping another player), the offending player must sit in the penalty box, and his team must play with one fewer player on the ice. The penalized team is said to be "short-handed", while the other team has a "powerplay". If a player scores while his team is short handed, this is recorded as a short-handed goal.
Patrick Marleau, 4
Marian Hossa, 3
Brian Gionta, 3
Jarome Iginla, 3
Mike Fisher, 3
Dave Bolland, 3
Jonathan Toews, 3
Playoff game-winning goals (active)[edit]
Jaromir Jagr, 16
Patrick Marleau, 16
Marian Hossa, 12
Johan Franzen, 12
Patrick Kane, 11
Joe Pavelski, 11
Jonathan Toews, 10
Evgeni Malkin, 10
Sidney Crosby, 9
Henrik Zetterberg, 8
Brian Gionta, 8
Joe Thornton, 8
Pavel Datsyuk, 7
David Krejci, 7
Jarome Iginla, 7
Tyler Johnson, 7
Playoff overtime goals (active)[edit]
If a game is tied after regulation time (which lasts three 20-minutes periods), there will be a period of "overtime" to decide the winner. The player who scores during this extra time is given the overtime goal. All overtime in the NHL is sudden death—meaning the first team to score is the winner—so the player who scores in overtime also has the game-winning goal.
Patrick Kane, 5
Jaromir Jagr, 4
Patrick Marleau, 4
Patrice Bergeron, 3
Alexandre Burrows, 3
Joe Pavelski, 3
Devin Setoguchi, 3
Nick Bonino, 3
Nicklas Backstrom, 3
Brent Seabrook, 3
Playoff assists (active)[edit]
Further information: Assist (ice hockey)
Jaromir Jagr, 123
Sidney Crosby, 107
Evgeni Malkin, 99
Marian Hossa, 97
Joe Thornton, 96
Ryan Getzlaf, 81
Patrick Kane, 73
Jonathan Toews, 70
Chris Kunitz, 65
Henrik Zetterberg, 63,
Duncan Keith, 63
Justin Williams, 58
Valtteri Filppula, 55,
Henrik Sedin, 55
Corey Perry, 53
Patrick Marleau, 52
Kris Letang, 50
Nicklas Backstrom, 49
David Krejci, 48
Daniel Sedin, 46
Playoff games played (active)[edit]
Further information: Games played
Jaromir Jagr, 208
Marian Hossa, 205
Patrick Marleau, 177
Chris Kunitz, 161
Joe Thornton, 160
Valtteri Filppula, 152
Evgeni Malkin, 149
Sidney Crosby, 148
Zdeno Chara, 147
Patrick Sharp, 142
Playoff penalty minutes (active)[edit]
A penalty is given to a player for committing an infraction during the game. The length of the penalty varies depending on the severity of the offence. The amount of penalty minutes recorded for statistical purposes are:
minor – 2 minutes
double minor – 4 minutes
major – 5 minutes
misconduct – 10 minutes
game misconduct – 10 minutes
Chris Neil, 192
Zdeno Chara, 186
Jaromir Jagr, 159
Milan Lucic, 155
Scott Hartnell, 146
Playoff plus-minus (active)[edit]
Plus-minus is a statistic that indicates the relative goal differential when a player is on the ice. If the player is on the ice when his team scores even-strength or short-handed, he is given +1; if he is on the ice when the opposing team scores even-strength or short-handed, he is given -1.
Henrik Zetterberg, +41
Milan Lucic, +38
Duncan Keith, +37
Zdeno Chara, +37
Nathan Horton, +31
Jaromir Jagr, +30
Patrice Bergeron, +30
Marian Hossa, +29
Justin Williams, +29
Marian Hossa, +29
David Krejci, +28
Niklas Hjalmarsson, +26
Johan Franzen, +26
Niklas Kronwall, +25
Anze Kopitar, +24
Bryan Bickell, +21
Playoff shots on goal (active)[edit]
Further information: Shot on goal
Marian Hossa, 697
Jaromir Jagr, 682
Henrik Zetterberg, 514
Evgeni Malkin, 463
Patrick Marleau, 445
Playoff shooting percentage (active)[edit]
Shooting percentage is the percentage of shots on goal which result in a goal.
Minimum 80 shots
Tyler Johnson, 20.2%
Nikita Kucherov, 19.8%
Vladimir Tarasenko, 19.4%
Jussi Jokinen, 19.1%
Chris Kreider, 17.5%
Mike Cammalleri, 17.0%
Ondrej Palat, 16.9%
Nathan Horton, 16.7%
Claude Giroux, 16.5%
Joel Ward, 15.7%
Goaltenders[edit]
The statistics listed include the 2015–16 NHL regular season and 2016 playoffs.
All-time leaders (goaltenders)[edit]
Active goaltenders (during 2015–16 NHL season) are listed in boldface.
Regular season wins[edit]
Martin Brodeur, 691
Patrick Roy, 551
Ed Belfour, 484
Curtis Joseph, 454
Roberto Luongo, 453
Terry Sawchuk, 447
Jacques Plante, 437
Tony Esposito, 423
Glenn Hall, 407
Grant Fuhr, 403
Henrik Lundqvist, 405
Chris Osgood, 401
Dominik Hasek, 389
Mike Vernon, 385
Marc-Andre Fleury, 375
John Vanbiesbrouck, 374
Tom Barrasso, 369
Ryan Miller, 358
Rogie Vachon, 355
Evgeni Nabokov, 353
Gump Worsley, 335
Nikolai Khabibulin, 333
Harry Lumley, 330
Sean Burke, 324
Miika Kiprusoff, 319
Regular season shutouts[edit]
A goaltender achieves a shutout when he does not allow a goal against him, and plays the full game.
Martin Brodeur, 125
Terry Sawchuk, 103
George Hainsworth, 94
Glenn Hall, 84
Jacques Plante, 82
Dominik Hasek, 81
Alex Connell, 81
Tiny Thompson, 81
Ed Belfour, 76
Tony Esposito, 76
Roberto Luongo, 73
Harry Lumley, 71
Roy Worters, 67
Patrick Roy, 66
Turk Broda, 62
Henrik Lundqvist, 61
Evgeni Nabokov, 59
John Ross Roach, 58
Clint Benedict, 57
Eddie Giacomin, 54
Bernie Parent, 54
Curtis Joseph, 51
Dave Kerr, 51
Rogie Vachon, 51
Tomas Vokoun, 51
Regular season goals against average[edit]
Goals against average is the average number of goals a goaltender allows over a 60-minute period (the regulation length of a game). It is calculated by multiplying the goals against by 60 minutes, then dividing by the total minutes played.
Minimum 250 games played
Alec Connell, 1.912
George Hainsworth, 1.933
Chuck Gardiner, 2.024
Lorne Chabot, 2.039
Tiny Thompson, 2.077
Dave Kerr, 2.149
Dominik Hasek, 2.202
Martin Brodeur, 2.230
Ken Dryden, 2.235
Roy Worters, 2.273
Henrik Lundqvist, 2.280
Roman Turek, 2.306
Clint Benedict, 2.315
Pekka Rinne, 2.350
Gerry McNeil, 2.355
Bill Durnan, 2.356
Marty Turco, 2.357
Jacques Plante, 2.379
Evgeni Nabokov, 2.400
Manny Legace, 2.410
Niklas Backstrom, 2.422
Carey Price, 2.440
Miikka Kiprusoff, 2.452
John Ross Roach, 2.456
Cristobal Huet, 2.457
Regular season save percentage[edit]
Save percentage is the percentage of shots on goal that a goaltender stops. It is calculated by dividing the number of saves by the number of shots on goal.
Minimum 250 games played
Dominik Hasek, .9223
Henrik Lundqvist, .9215
Tim Thomas, .9214
Pekka Rinne, .9195
Roberto Luongo, .9193
Jonas Hiller, .9178
Niklas Backstrom, .9177
Tomas Vokoun, .9168
Carey Price, .9160
Ryan Miller, .91474
Ilya Bryzgalov, .91471
Jonathan Quick, .9146
Kari Lehtonen, .9141
Miikka Kiprusoff, .9136
Cristobal Huet, .9134
Craig Anderson, .9133
Mike Smith, .9132
Martin Brodeur, .91299
Jean-Sebastien Giguere, .91297
Evgeni Nabokov, .9124
Manny Fernandez, .9120
Manny Legace, .9116
Cam Ward, .9104
Martin Biron, .9102
Patrick Roy, .9102
Playoff wins[edit]
Patrick Roy, 151
Martin Brodeur, 113
Grant Fuhr, 92
Ed Belfour, 88
Billy Smith, 88
Ken Dryden, 80
Mike Vernon, 77
Chris Osgood, 74
Jacques Plante, 71
Andy Moog, 68
Dominik Hasek, 65
Curtis Joseph, 63
Tom Barrasso, 61
Turk Broda, 60
Henrik Lundqvist, 55
Terry Sawchuk, 54
Gerry Cheevers, 53
Marc-Andre Fleury, 52
Glenn Hall, 49
Corey Crawford, 48
Playoff shutouts[edit]
A goaltender achieves a shutout when he does not allow a goal against him, and plays the full game.
Martin Brodeur, 24
Patrick Roy, 23
Curtis Joseph, 16
Chris Osgood, 15
Ed Belfour, 14
Dominik Hasek, 14
Jacques Plante, 14
Turk Broda, 13
Terry Sawchuk, 12
Ken Dryden, 10
Henrik Lundqvist, 9
Jonathan Quick, 9
Clint Benedict, 9
Mike Richter, 9
Gerry Cheevers, 8
George Hainsworth, 8
Dave Kerr, 8
Felix Potvin, 8
Marc-Andre Fleury, 8
Harry Lumley, 7
Evgeni Nabokov, 7
John Ross Roach, 7
Tiny Thompson, 7
Active leaders (goaltenders)[edit]
Regular season wins (active)[edit]
Roberto Luongo, 436
Henrik Lundqvist, 374
Marc-Andre Fleury, 357
Ryan Miller, 340
Kari Lehtonen, 273
Cam Ward, 269
Jonathan Quick, 252
Pekka Rinne, 238
Carey Price, 233
Antti Niemi, 215
Regular season shutouts (active)[edit]
A goaltender achieves a shutout when he does not allow a goal against him, and plays the full game.
Roberto Luongo, 72
Henrik Lundqvist, 59
Marc-Andre Fleury, 43
Jonathan Quick, 42
Pekka Rinne, 40
Jaroslav Halak, 39
Ryan Miller, 36
Carey Price, 36
Antti Niemi, 35
Brian Elliott, 34
Kari Lehtonen, 34
Regular season goals against average (active)[edit]
Goals against average is the average number of goals a goaltender allows over a 60-minute period (the regulation length of a game). It is calculated by multiplying the goals against by 60 minutes, then dividing by the total minutes played.
Minimum 250 games played
Cory Schneider, 2.16
Tuukka Rask, 2.24
Jonathan Quick, 2.27
Henrik Lundqvist, 2.28
Corey Crawford, 2.34
Pekka Rinne, 2.37
Jaroslav Halak, 2.38
Brian Elliott, 2.40
Regular season save percentage (active)[edit]
Save percentage is the percentage of shots on goal that a goaltender stops. It is calculated by dividing the number of saves by the number of shots on goal.
Minimum 250 games played
Cory Schneider, .9250
Tuukka Rask, .9241
Henrik Lundqvist, .9205
Carey Price, .9197
Roberto Luongo, .9192
Corey Crawford, .9179
Pekka Rinne, .9173
Semyon Varlamov, .9172
Jaroslav Halak, .9172
Sergei Bobrovsky, .9170
Jonathan Quick, .9157
Craig Anderson, .9154
Playoff wins (active)[edit]
Henrik Lundqvist, 55
Marc-Andre Fleury, 53
Corey Crawford, 48
Jonathan Quick, 46
Antti Niemi, 36
Roberto Luongo, 34
Ryan Miller, 28
Tuukka Rask, 28
Cam Ward, 23
Carey Price, 23
Playoff shutouts (active)[edit]
A goaltender achieves a shutout when he does not allow a goal against him, and plays the full game.
Henrik Lundqvist, 9
Jonathan Quick, 9
Marc-Andre Fleury, 8
Corey Crawford, 5
Roberto Luongo, 5
Carey Price, 5
Tuukka Rask, 5
Ben Bishop, 5
Cam Ward, 4
Braden Holtby, 4
Mike Smith, 3
Jimmy Howard, 3
Petr Mrazek, 3
Craig Anderson, 3
Michael Leighton, 3
Martin Jones, 3
Ryan Miller, 3
Coaches[edit]
The statistics listed include the 2015–16 NHL season.
All-time leaders (coaches)[edit]
Active coaches (during 2015–16 NHL season) are listed in boldface.
Regular season games coached[edit]
Scotty Bowman, 2,141
Al Arbour, 1,607
Joel Quenneville, 1,481
Dick Irvin, Sr., 1,449
Ron Wilson, 1,401
Pat Quinn, 1,400
Mike Keenan, 1,386
Lindy Ruff, 1,329
Ken Hitchcock, 1,322
Jacques Martin, 1,294
Barry Trotz, 1,278
Jacques Lemaire, 1,262
Bryan Murray, 1,239
Paul Maurice, 1,201
Marc Crawford, 1,151
Darryl Sutter, 1,121
Billy Reay, 1,102
Brian Sutter, 1,028
Pat Burns, 1,019
Terry Murray, 1,012
Regular season coaching wins[edit]
Scotty Bowman, 1,244
Joel Quenneville, 801
Al Arbour, 782
Ken Hitchcock, 781
Dick Irvin, Sr., 692
Pat Quinn, 684
Mike Keenan, 672
Lindy Ruff, 652
Ron Wilson, 648
Bryan Murray, 620
Jacques Lemaire, 617
Jacques Martin, 613
Barry Trotz, 602
Marc Crawford, 549
Darryl Sutter, 547
Billy Reay, 542
Mike Babcock, 527
Paul Maurice, 521
Alain Vigneault, 520
Pat Burns, 501
Regular season coaching points percentage[edit]
Points percentage is determined by the number of points a team earns (equal to the number of ties and overtime losses, plus twice the number of wins) divided by the total possible points (equal to twice the number of games).
Minimum 200 games coached
Tom Johnson, .738
Dan Bylsma, .668
Bruce Boudreau, .663
Scotty Bowman, .657
Claude Ruel, .648
Todd McLellan, .637
Toe Blake, .634
Mike Babcock, .627
Floyd Smith, .626
Joel Quenneville, .614
Fred Shero, .612
Claude Julien, .611
Ken Hitchcock, .605
Gerry Cheevers, .6037
Dave Lewis, .6036
Glen Sather, .602
Don Cherry, .601
Mike Sullivan, .600
Tommy Ivan, .599
Alain Vigneault, .592
Jimmy Skinner, .591
Playoff games coached[edit]
Scotty Bowman, 353
Al Arbour, 209
Joel Quenneville, 204
Dick Irvin, Sr., 190
Pat Quinn, 183
Mike Keenan, 173
Darryl Sutter, 165
Pat Burns, 149
Ken Hitchcock, 148
Mike Babcock, 144
Glen Sather, 127
Alain Vigneault, 122
Toe Blake, 119
Jacques Lemaire, 117
Billy Reay, 117
Playoff coaching wins[edit]
Scotty Bowman, 223
Al Arbour, 123
Joel Quenneville, 115
Dick Irvin, 100
Mike Keenan, 96
Pat Quinn, 94
Glen Sather, 89
Darryl Sutter, 88
Toe Blake, 82
Mike Babcock, 82
Pat Burns, 78
Ken Hitchcock, 76
Fred Shero, 63
Jacques Lemaire, 61
Claude Julien, 61
Alain Vigneault, 61
Playoff coaching win percentage[edit]
Minimum 25 games coached
Glen Sather, .705
Toe Blake, .689
Tom Johnson, .682
Claude Ruel, .667
Scotty Bowman, .632
Mike Sullivan, .625
Jean Perron, .625
Hap Day, .613
Tommy Gorman, .600
Larry Robinson, .596
Al Arbour, .589
Mike Milbury, .575
Fred Shero, .573
Mike Babcock, .569
Bob Hartley, .568
Craig MacTavish, .567
Stanley Cups[edit]
Scotty Bowman, 9
Toe Blake, 8
Hap Day, 5
Al Arbour, 4
Punch Imlach, 4
Dick Irvin, 4
Glen Sather, 4
Jack Adams, 3
Pete Green, 3
Tommy Ivan, 3
Joel Quenneville, 3
Active leaders (coaches)[edit]
Active Leaders section updated to end of 2012-13 season, not 2014-15 season like the All Time section above.
Regular season games coached (active)[edit]
Joel Quenneville, 1,211
Lindy Ruff, 1,165
Ken Hitchcock, 1,158
Barry Trotz, 1,114
Paul Maurice, 1,119
Darryl Sutter, 957
Alain Vigneault, 806
Mike Babcock, 786
Dave Tippett, 786
Bob Hartley, 698
Regular season coaching wins (active)[edit]
Joel Quenneville, 801
Ken Hitchcock, 700
Lindy Ruff, 612
Barry Trotz, 503
Mike Babcock, 500
Paul Maurice, 478
Darryl Sutter, 434
Dave Tippett, 406
Alain Vigneault, 396
Peter Laviolette, 366
Regular season coaching points percentage (active)[edit]
Points percentage is determined by the number of points a team earns (equal to the number of ties and overtime losses, plus twice the number of wins) divided by the total possible points (equal to twice the number of games).
Minimum 200 games coached
Todd McLellan, .657
Bruce Boudreau, .651
Mike Babcock, .639
Dave Tippett, .617
Joel Quenneville, .604
Playoff games coached (active)[edit]
Joel Quenneville, 163
Darryl Sutter, 139
Ken Hitchcock, 136
Mike Babcock, 131
Lindy Ruff, 101
Playoff coaching wins (active)[edit]
Joel Quenneville, 107
Mike Babcock, 78
Ken Hitchcock, 72
Darryl Sutter, 72
Lindy Ruff, 57
Stanley Cups (active)[edit]
Joel Quenneville, 3
Mike Sullivan, 2
Darryl Sutter, 2
Mike Babcock, 1
Dan Bylsma, 1
Ken Hitchcock, 1
Claude Julien, 1
Peter Laviolette, 1
John Tortorella, 1
Notes[edit]
Jump up ^ A player is not eligible for induction into the Hockey Hall of Fame until he has been fully retired for three calendar years or is deceased; player was not eligible for induction upon the commencement of the 2014-15 NHL season.
External links[edit]
Career player statistics tables from NHL.com
Leaders and Records Index from hockey-reference.com
NHL Coach Register from hockey-reference.com
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text:
1 + 2 + 3 + 4 + ⋯
From Wikipedia, the free encyclopedia
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The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯. The parabola is their smoothed asymptote; its y-intercept is − 1/12.[1]
The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number
∑ k = 1 n k = n ( n + 1 ) 2 , {\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},}
which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.
Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results, some of which have applications in other fields such as complex analysis[citation needed], quantum field theory[citation needed], and string theory. Many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of − 1/12, which is expressed by a famous formula:[2]
1 + 2 + 3 + 4 + ⋯ = − 1 12 , {\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}
where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning.
In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".[3]
Contents
[hide]
1 Partial sums
2 Summability
2.1 Heuristics
2.2 Zeta function regularization
2.3 Cutoff regularization
2.4 Ramanujan summation
2.5 Failure of stable linear summation methods
3 Physics
4 History
5 In popular media
6 Notes
7 References
8 Further reading
9 External links
Partial sums[edit]
The first six triangular numbers
Main article: Triangular number
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc. The nth partial sum is given by a simple formula:
∑ k = 1 n k = n ( n + 1 ) 2 . {\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.}
This equation was known to the Pythagoreans as early as the sixth century BCE.[4] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test.
Summability[edit]
Among the classical divergent series, 1 + 2 + 3 + 4 + ⋯ is relatively difficult to manipulate into a finite value. Many summation methods are used to assign numerical values to divergent series, some more powerful than others. For example, Cesàro summation is a well-known method that sums Grandi's series, the mildly divergent series 1 − 1 + 1 − 1 + ⋯, to 1⁄2. Abel summation is a more powerful method that not only sums Grandi's series to 1⁄2, but also sums the trickier series 1 − 2 + 3 − 4 + ⋯ to 1⁄4.
Unlike the above series, 1 + 2 + 3 + 4 + ⋯ is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞.[5] Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value; see below. More advanced methods are required, such as zeta function regularization or Ramanujan summation. It is also possible to argue for the value of − 1/12 using some rough heuristics related to these methods.
Heuristics[edit]
Passage from Ramanujan's first notebook describing the "constant" of the series
Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = − 1/12" in chapter 8 of his first notebook.[6][7][8] The simpler, less rigorous derivation proceeds in two steps, as follows.
The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + ⋯ closely resembles the alternating series 1 − 2 + 3 − 4 + ⋯. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.[9]
In order to transform the series 1 + 2 + 3 + 4 + ⋯ into 1 − 2 + 3 − 4 + ⋯, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯. Then multiply this equation by 4 and subtract the second equation from the first:
c = 1 + 2 + 3 + 4 + 5 + 6 + ⋯ 4 c = 4 + 8 + 12 + ⋯ c − 4 c = 1 − 2 + 3 − 4 + 5 − 6 + ⋯ {\displaystyle {\begin{alignedat}{7}c&{}={}&1+2&&{}+3+4&&{}+5+6+\cdots \\4c&{}={}&4&&{}+8&&{}+12+\cdots \\c-4c&{}={}&1-2&&{}+3-4&&{}+5-6+\cdots \\\end{alignedat}}}
The second key insight is that the alternating series 1 − 2 + 3 − 4 + ⋯ is the formal power series expansion of the function 1/(1 + x)2 but with x defined as 1. Accordingly, Ramanujan writes:
− 3 c = 1 − 2 + 3 − 4 + ⋯ = 1 ( 1 + 1 ) 2 = 1 4 {\displaystyle -3c=1-2+3-4+\cdots ={\frac {1}{(1+1)^{2}}}={\frac {1}{4}}}
Dividing both sides by −3, one gets c = − 1/12.
Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. For an extreme example, appending a single zero to the front of the series can lead to inconsistent results.[1]
One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.[10] In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to a function n−s, where s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be set to −1 later. The implementation of this strategy is called zeta function regularization.
Zeta function regularization[edit]
Plot of ζ(s). For s > 1, the series converges and ζ(s) > 1. Analytic continuation around the pole at s = 1 leads to a region of negative values, including ζ(−1) = − 1/12
In zeta function regularization, the series ∑ n = 1 ∞ n {\displaystyle \sum _{n=1}^{\infty }n} is replaced by the series ∑ n = 1 ∞ n − s {\displaystyle \sum _{n=1}^{\infty }n^{-s}} . The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s). On the other hand, the Dirichlet series diverges when the real part of s is less than or equal to 1, so, in particular, the series 1 + 2 + 3 + 4 + ⋯ that results from setting s = –1 does not converge. The benefit of introducing the Riemann zeta function is that it can be defined for other values of s by analytic continuation. One can then define the zeta-regularized sum of 1 + 2 + 3 + 4 + ⋯ to be ζ(−1).
From this point, there are a few ways to prove that ζ(−1) = − 1/12. One method, along the lines of Euler's reasoning,[11] uses the relationship between the Riemann zeta function and the Dirichlet eta function η(s). The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities:
ζ ( s ) = 1 − s + 2 − s + 3 − s + 4 − s + 5 − s + 6 − s + ⋯ 2 × 2 − s ζ ( s ) = 2 × 2 − s + 2 × 4 − s + 2 × 6 − s + ⋯ ( 1 − 2 1 − s ) ζ ( s ) = 1 − s − 2 − s + 3 − s − 4 − s + 5 − s − 6 − s + ⋯ = η ( s ) {\displaystyle {\begin{alignedat}{7}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\times 2^{-s}\zeta (s)&{}={}&2\times 2^{-s}&&{}+2\times 4^{-s}&&{}+2\times 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s)\\\end{alignedat}}}
The identity ( 1 − 2 1 − s ) ζ ( s ) = η ( s ) {\displaystyle (1-2^{1-s})\zeta (s)=\eta (s)} continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. Substituting s = −1, one gets −3ζ(−1) = η(−1). Now, computing η(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series,[12] which is a one-sided limit:
− 3 ζ ( − 1 ) = η ( − 1 ) = lim x → 1 − ( 1 − 2 x + 3 x 2 − 4 x 3 + ⋯ ) = lim x → 1 − 1 ( 1 + x ) 2 = 1 4 {\displaystyle -3\zeta (-1)=\eta (-1)=\lim _{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots \right)=\lim _{x\to 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}}
Dividing both sides by −3, one gets ζ(−1) = − 1/12.
Cutoff regularization[edit]
The series 1 + 2 + 3 + 4 + ⋯
After smoothing
Asymptotic behavior of the smoothing. The y-intercept of the parabola is − 1/12.[1]
The method of regularization using a cutoff function can "smooth" the series to arrive at − 1/12. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from real analysis.
The idea is to replace the ill-behaved discrete series ∑ n = 0 N n {\displaystyle \sum _{n=0}^{N}n} with a smoothed version
∑ n = 0 ∞ n f ( n N ) {\displaystyle \sum _{n=0}^{\infty }nf\left({\frac {n}{N}}\right)} ,
where f is a cutoff function with appropriate properties. The cutoff function must be normalized to f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require that f is smooth, bounded, and compactly supported. One can then prove that this smoothed sum is asymptotic to − 1/12 + CN2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic continuation, − 1/12.[1]
Ramanujan summation[edit]
The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also − 1/12. Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ = − 1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"[13]
Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f, the classical Ramanujan sum of the series ∑ k = 1 ∞ f ( k ) {\displaystyle \sum _{k=1}^{\infty }f(k)} is defined as
c = − 1 2 f ( 0 ) − ∑ k = 1 ∞ B 2 k ( 2 k ) ! f ( 2 k − 1 ) ( 0 ) , {\displaystyle c=-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0),}
where f(2k−1) is the (2k − 1)-th derivative of f and B2k is the 2kth Bernoulli number: B2 = 1/6, B4 = − 1/30, and so on. Setting f(x) = x, the first derivative of f is 1, and every other term vanishes, so:[14]
c = − 1 6 × 1 2 ! = − 1 12 . {\displaystyle c=-{\frac {1}{6}}\times {\frac {1}{2!}}=-{\frac {1}{12}}.}
To avoid inconsistencies, the modern theory of Ramanujan summation requires that f is "regular" in the sense that the higher-order derivatives of f decay quickly enough for the remainder terms in the Euler–Maclaurin formula to tend to 0. Ramanujan tacitly assumed this property.[14] The regularity requirement prevents the use of Ramanujan summation upon spaced-out series like 0 + 2 + 0 + 4 + ⋯, because no regular function takes those values. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series.[15]
Failure of stable linear summation methods[edit]
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A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term to the beginning of the series increases the sum by the same amount.) This can be seen as follows. If
1 + 2 + 3 + ⋯ = x
then adding 0 to both sides gives
0 + 1 + 2 + ⋯ = 0 + x = x by stability.
By linearity, one may subtract the second equation from the first (subtracting each component of the second line from the first line in columns) to give
1 + 1 + 1 + ⋯ = x – x = 0.
Adding 0 to both sides again gives
0 + 1 + 1 + 1 + ⋯ = 0,
and subtracting the last two series gives
1 + 0 + 0 + ⋯ = 0
contradicting stability.
The methods used above to sum 1 + 2 + 3 + ⋯ are either not stable or not linear.
Physics[edit]
In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particular the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of D − 2 independent quantum harmonic oscillators, one for each transverse direction, where D is the dimension of spacetime. If the fundamental oscillation frequency is ω then the energy in an oscillator contributing to the nth harmonic is nħω/2. So using the divergent series, the sum over all harmonics is − ħω(D − 2)/24. Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.[16]
The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension.[17] An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates. The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term − 1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.[18]
A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.[19]
History[edit]
It is unclear whether Leonhard Euler summed the series to − 1/12. According to Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + ⋯ = ∞.[20] According to Raymond Ayoub, the fact that the divergent zeta series is not Abel summable prevented Euler from using the zeta function as freely as the eta function, and he "could not have attached a meaning" to the series.[21] Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta and eta functions to negative integers.[22][23][24] In the primary literature, the series 1 + 2 + 3 + 4 + ⋯ is mentioned in Euler's 1760 publication De seriebus divergentibus alongside the divergent geometric series 1 + 2 + 4 + 8 + ⋯. Euler hints that series of this type have finite, negative sums, and he explains what this means for geometric series, but he does not return to discuss 1 + 2 + 3 + 4 + ⋯. In the same publication, Euler writes that the sum of 1 + 1 + 1 + 1 + ⋯ is infinite.[25]
In popular media[edit]
David Leavitt's 2007 novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series. They conclude that Ramanujan has rediscovered ζ(−1), and they take the "lunatic asylum" line in his second letter as a sign that Ramanujan is toying with them.[26]
Simon McBurney's 2007 play A Disappearing Number focuses on the series in the opening scene. The main character, Ruth, walks into a lecture hall and introduces the idea of a divergent series before proclaiming, "I'm going to show you something really thrilling," namely 1 + 2 + 3 + 4 + ⋯ = − 1/12. As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is real. It's terrifying, but it's real."[27][28]
In January 2014, Numberphile produced a YouTube video on the series, which gathered over 1.5 million views in its first month.[29] The 8-minute video is narrated by Tony Padilla, a physicist at the University of Nottingham. Padilla begins with 1 − 1 + 1 − 1 + ⋯ and 1 − 2 + 3 − 4 + ⋯ and relates the latter to 1 + 2 + 3 + 4 + ⋯ using a term-by-term subtraction similar to Ramanujan's argument.[30] Numberphile also released a 21-minute version of the video featuring Nottingham physicist Ed Copeland, who describes in more detail how 1 − 2 + 3 − 4 + ⋯ = 1/4 as an Abel sum and 1 + 2 + 3 + 4 + ⋯ = − 1/12 as ζ(−1).[31] After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.[32]
In the New York Times coverage of the Numberphile video, mathematician Edward Frenkel commented, "This calculation is one of the best-kept secrets in math. No one on the outside knows about it."[29]
Coverage of this topic in Smithsonian magazine describes the Numberphile video as misleading, and notes that the interpretation of the sum as − 1/12 relies on a specialized meaning for the equals sign, from the techniques of analytic continuation, in which equals means is associated with.[33]
Notes[edit]
^ Jump up to: a b c d Tao, Terence (April 10, 2010), The Euler–Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
Jump up ^ Lepowsky, J. (1999), Naihuan Jing and Kailash C. Misra, ed., Vertex operator algebras and the zeta function, Contemporary Mathematics, 248, pp. 327–340, arXiv:math/9909178  , Bibcode:1999math......9178L
Jump up ^ Gannon, Terry (April 2010), Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, p. 140, ISBN 978-0521141888
Jump up ^ Pengelley, David J. (2002), Otto Bekken; et al., eds., The bridge between the continuous and the discrete via original sources, National Center for Mathematics Education, University of Gothenburg, Sweden, p. 3, ISBN 978-9185143009
Jump up ^ Hardy p.10
Jump up ^ Ramanujan's Notebooks, retrieved January 26, 2014
Jump up ^ Abdi, Wazir Hasan (1992), Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician, National, p. 41
Jump up ^ Berndt, Bruce C. (1985), Ramanujan's Notebooks: Part 1, Springer-Verlag, pp. 135–136
Jump up ^ Euler, Leonhard (2006). "Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series". Translated by Willis, Lucas; Osler, Thomas J. The Euler Archive. Retrieved 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Mémoires de l'Académie des Sciences de Berlin. 17: 83–106.
Jump up ^ Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 475–476. ISBN 0-486-66165-2.
Jump up ^ Stopple, Jeffrey (2003), A Primer of Analytic Number Theory: From Pythagoras to Riemann, p. 202, ISBN 0-521-81309-3
Jump up ^ Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 490–492. ISBN 0-486-66165-2.
Jump up ^ Berndt et al. p.53.
^ Jump up to: a b Berndt, Bruce C. (1985), Ramanujan's Notebooks: Part 1, Springer-Verlag, pp. 13, 134
Jump up ^ Hardy p.346
Jump up ^ Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories", Physics Letters B, 190 (1–2): 137–139, Bibcode:1987PhLB..190..137B, doi:10.1016/0370-2693(87)90854-9
Jump up ^ See v:Quantum mechanics/Casimir effect in one dimension
Jump up ^ Zee pp.65–67
Jump up ^ Zeidler, Eberhard (2007), "Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists", Quantum Field Theory I: Basics in Mathematics and Physics. A bridge between mathematicians and physicists. by Eberhard Zeidler. ISBN 978-3-540-34762-0. Published by Springer Science+Business Media, Springer: 305–306, Bibcode:2006qftb.book.....Z, ISBN 9783540347644
Jump up ^ Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–314, doi:10.2307/2690371, JSTOR 2690371
Jump up ^ Ayoub, Raymond (December 1974), "Euler and the Zeta Function" (PDF), The American Mathematical Monthly, 81 (10): 1067–1086, doi:10.2307/2319041, JSTOR 2319041, retrieved February 14, 2014
Jump up ^ Lefort, Jean, "Les séries divergentes chez Euler" (PDF), L'Ouvert, IREM de Strasbourg (31): 15–25, archived from the original (PDF) on February 22, 2014, retrieved February 14, 2014
Jump up ^ Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171  , doi:10.2206/kyushujm.57.175, retrieved January 31, 2014
Jump up ^ Sondow, Jonathan (February 1994), "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proceedings of the American Mathematical Society, 120 (4): 421–424, doi:10.1090/S0002-9939-1994-1172954-7, retrieved February 14, 2014
Jump up ^ Barbeau, E.J.; Leah, P.J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6
Jump up ^ Leavitt, David (2007), The Indian Clerk, Bloomsbury, pp. 61–62
Jump up ^ Complicite (April 2012), A Disappearing Number, Oberon, ISBN 9781849432993
Jump up ^ Thomas, Rachel (December 1, 2008), "A disappearing number", Plus, retrieved February 5, 2014
^ Jump up to: a b Overbye, Dennis (February 3, 2014), "In the End, It All Adds Up to –1/12", New York TImes, retrieved February 3, 2014
Jump up ^ ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = –1/12 on YouTube
Jump up ^ Sum of Natural Numbers (second proof and extra footage) on YouTube
Jump up ^ Padilla, Tony, What do we get if we sum all the natural numbers?, retrieved February 3, 2014
Jump up ^ Schultz, Colin (2014-01-31). "The Great Debate Over Whether 1+2+3+4..+ ∞ = −1/12". Smithsonian. Retrieved 2016-05-16.
References[edit]
Berndt, Bruce C.; Srinivasa Ramanujan Aiyangar; Rankin, Robert A. (1995). Ramanujan: letters and commentary. American Mathematical Society. ISBN 0-8218-0287-9.
Hardy, G. H. (1949). Divergent Series. Clarendon Press.
Zee, A. (2003). Quantum field theory in a nutshell. Princeton UP. ISBN 0-691-01019-6.
Further reading[edit]
Lepowsky, James (1999). "Vertex operator algebras and the zeta function". Contemporary Mathematics. Contemporary Mathematics. 248: 327–340. arXiv:math/9909178  . doi:10.1090/conm/248/03829. ISBN 9780821811993.
Zwiebach, Barton (2004). A First Course in String Theory. Cambridge UP. ISBN 0-521-83143-1. See p. 293.
Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076  .
Watson, G. N. (April 1929), "Theorems stated by Ramanujan (VIII): Theorems on Divergent Series", Journal of the London Mathematical Society, 1, 4 (2): 82–86, doi:10.1112/jlms/s1-4.14.82
External links[edit]
Wikiversity has learning resources about divergent series
Lamb E. (2014), "Does 1+2+3… Really Equal –1/12?", Scientific American Blogs.
This Week's Finds in Mathematical Physics (Week 124), (Week 126), (Week 147), (Week 213)
Euler's Proof That 1 + 2 + 3 + ⋯ = −1/12 – by John Baez
García Moreta, José Javier http://prespacetime.com/index.php/pst/article/view/498 The Application of Zeta Regularization Method to the Calculation of Certain Divergent Series and Integrals Refined Higgs, CMB from Planck, Departures in Logic, and GR Issues & Solutions vol 4 Nº 3 prespacetime journal http://prespacetime.com/index.php/pst/issue/view/41/showToc
John Baez (September 19, 2008). "My Favorite Numbers: 24" (PDF).
The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation by Terence Tao
A recursive evaluation of zeta of negative integers by Luboš Motl
Link to Numberphile video 1 + 2 + 3 + 4 + 5 + ... = –1/12
Sum of Natural Numbers (second proof and extra footage) includes demonstration of Euler's method.
What do we get if we sum all the natural numbers? response to comments about video by Tony Padilla
Related article from New York Times
Why –1/12 is a gold nugget follow-up Numberphile video with Edward Frenkel
Divergent Series: why 1 + 2 + 3 + ⋯ = −1/12 by Brydon Cais from University of Arizona
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