Zeta Notatios Traditioal ame Riema zeta fuctio Traditioal otatio Ζs Mathematica StadardForm otatio Zetas Primary defiitio... Ζs ; Res s k k Specific values Specialized values..3.. Ζ B ;..3.. Ζ ;..3.3. Ζ Π B ;..3.53. Ζ Π k k k r r r k r ;..3.4. Ζ Π E ;
http://fuctios.wolfram.com Values at fixed poits..3.5. Ζ..3.6. Ζ9 3..3.7. Ζ8..3.8. Ζ7 4..3.9. Ζ6..3.. Ζ5 5..3.. Ζ4..3.. Ζ3..3.3. Ζ..3.4. Ζ..3.5. Ζ..3.6. Ζ Ζ Π 6..3.7...3.8. Ζ3 7Π3 8 k k 3 Π k..3.54. k 5 k 6 k 3 k Ζ3 k k 5 k
http://fuctios.wolfram.com 3..3.55. k 5 k 5 k 77 k Ζ3 64 k k 5 As of 3, the values of Ζ3 was computed with a accuracy of approximately decimal digits by usig above formula...3.56. Ζ3 3 Π 6 loga log Π Ψ 4..3.57. Ζ3 9 4 8 F 7,,,,,, Brychkov Yu.A. (6) Ζ4 Π4..3.9. 9 Ζ5 Π5..3.. 94 Ζ5 Π4 Ζ6 Π6 8 48 6, 7 35 k k 5 Π k 35 k k 5 Π k..3.58. 45 log Π Ψ6 6 Ψ 5 Ψ 4..3.. 945..3.. Ζ7 9Π7 56 7 k k 7 Π k Ζ7 Π6 Ζ8 945..3.59. 8 48 6 ; 4 3, 5 3, 3, 3,, 945 log Π 84 7 Ψ8 36 Ψ 7 6 Ψ 6 Ψ 4..3.3. Π8..3.4. Ζ9 5Π9 3 74 778..3.6. Π8 99 495 k k 9 Π k 495 k k 9 Π k 8 6, Ζ9 475 log Π 8 3 4 Ψ 5 Ψ 9 5 Ψ 8 4 Ψ 6 Ψ 4 8 6 ; 7
http://fuctios.wolfram.com 4 Ζ 93 555..3.5. Π..3.6. Ζ 453Π 45 675 5 k k Π k..3.7. 69 Π Ζ 638 5 875..3.8. Ζ3 89Π3 57 43 75..3.9. Π4 Ζ4 8 43 5..3.3. 6 5 855 k k 3 Π k 855 k k 3 Π k 3 687Π5 Ζ5 39 769 879 5 k k 5 Π k..3.3. 367 Π 6 Ζ6 35 64 566 5..3.3. 397 549Π 7 Ζ7 4 59 867 5..3.33. 43 867 Π 8 Ζ8 38 979 95 48 5..3.34. 6 63 3 85 k k 7 Π k 7 78 537Π 9 Ζ9 438 6 54 68 75 k k 9 Π k..3.35. 74 6 Π Ζ 53 39 465 9 65..3.36. 68 59 64 373Π Ζ 88 63 85 76 59 53 5..3.37. 55 366 Π Ζ 3 447 856 94 643 5 4 96 35 98 75 k k Π k 3 85 k k 7 Π k 98 75 k k Π k
http://fuctios.wolfram.com 5..3.38. 36 364 9 Π 4 Ζ4 99 57 963 756 5 875..3.39. 35 86 Π 6 Ζ6 94 48 976 3 578 5..3.4. 6 785 56 94 Π 8 Ζ8 564 653 66 7 76 73 67 875..3.4. 6 89 673 84 Π 3 Ζ3 5 66 878 84 669 8 674 7 5 65..3.4. 7 79 3 4 7 Π 3 Ζ3 6 49 57 34 7 66 46 5..3.43. 5 68 697 55 Π 34 Ζ34 3 454 58 433 748 587 9 89 65..3.44. 6 35 7 553 53 477 373 Π 36 Ζ36 777 977 56 866 588 586 487 68 66 44 9 875..3.45. 38 4 4 983 3 Π 38 Ζ38 43 467 68 49 375 776 343 76 883 984 375..3.46. 6 8 78 496 449 5 Π 4 Ζ4 8 43 7 89 638 86 798 4 8 39 556 64 65..3.47. 3 4 95 87 836 4 65 38 Π 4 Ζ4 37 789 89 88 96 7 7 594 47 864 667 47 734 375..3.48. 5 6 594 468 963 8 588 86 Π 44 Ζ44 37 93 679 547 5 773 56 76 98 457 776 679 69 9 875..3.49. 3 73 68 3 89 7 874 48 Π 46 Ζ46 7 67 4 448 3 53 33 358 48 6 59 46 89 65..3.5. 5 69 43 368 997 87 686 49 7 547 Π 48 Ζ48 4 93 648 63 384 74 996 59 698 9 478 879 58 6 86 669 9 875
http://fuctios.wolfram.com 6..3.5. 39 64 576 49 86 37 856 998 Π 5 Ζ5 85 58 77 457 546 764 463 363 635 5 374 44 83 54 365 34 375 Values at ifiities..3.5. Ζ Geeral characteristics Domai ad aalyticity Ζs is a aalytical fuctio of s, which is defied over the whole complex s-plae...4.. sζs Symmetries ad periodicities Mirror symmetry..4.. Ζs Ζs Periodicity No periodicity Poles ad essetial sigularities The fuctio Ζs has oly two sigular poits: a) s is the simple pole with residue ; b) s is a essetial sigular poit...4.3. ig s Ζs,,,..4.4. res s Ζs Brach poits The fuctio Ζs does ot have brach poits...4.5. s Ζs Brach cuts The fuctio Ζs does ot have brach cuts.
http://fuctios.wolfram.com 7..4.6. s Ζs Series represetatios Geeralized power series Expasios at s s ; s..6.7. Ζs Ζs Ζ s s s Ζ s s s ; s s s..6.8. Ζs Ζs Ζ s s s Ζ s s s Os s 3 ; s..6.9. Ζ k s s s k Ζs k k ; s..6.. Ζs Ζs Os s ; s Expasios at s..6.. Ζs log Π s log logπ..6.3. Ζs log Π s log logπ..6.. Ζs Os Γ Γ Π 4 s ; s Π 4 s Os 3 Expasios at s..6.3. Ζs s Γ s Γ s ; s..6.4. Ζs s Γ s Γ s Os 3..6.. Ζs k Γ s k s k k k
http://fuctios.wolfram.com 8..6.. Ζs Os s..6.5. Ζ s Ζs s Η k s k ; Η k k k k lim x x j log k j j j logk x k ; p e ; p e e a ; a e,e e a p e Krzysztof Maslaka..6.6. Ζ s Ζs s Η k s k ; k k j Η k k j c kj,j ; c,k k c m,k m m k m k i mi Γ mi c i,k j i m i Krzysztof Maslaka Expasios at s..6.3. Ζs 4 Π s Ζ log4 Π Ψ Ζ Ζ O s 3 ; s..6.4. Ζs Ζ Π s Os ; s Expoetial Fourier series..6.5. Ζs Π s s si Π s cos Π k cos Π s k k s si Π k ; Res k k s Asymptotic series expasios..6.6. Ζs Ζs ; s This meas it caot be represeted through other fuctios. Other series represetatios..6.7. Ζs ; Res s k k
http://fuctios.wolfram.com 9..6.8. Ζs k ; Res s s k k s..6.9. Ζs s k ; Res s k..6.. Ζs s k k k k s Ζs..6.7. k s k k s k Ζs..6.. s k..6.. t s k t t ; Res k t Ζs s s k k k k Ζ k Krzysztof Maslaka: Hypergeometric-like Represetatio of the Zeta-Fuctio of Riema math-ph/57 () http://arxiv.org/abs/math-ph/57 Krzysztof Maslaka: {}[[,]] Acta Cosmologica XIII-, {}[[,]] (997) For specialized values..6.6. m Ζ m m k k ; m m k m k..6.8. Ζ5 369 6 65 k 4 k 4 k 4 k 3 5 9 5 64 k 6 6 944 k 496 4 k 5 38 7 64 4 k 8 5 3 96 4 k 4 5 49 696 4 k 5 4 77 8 4 k 9 5 5 6 4 k 5 5 65 7 4 k 5 8 4 k 4 5 4 k 5 33 48 4 7 76 4 k 3 5 4 k 4 5 43 84 4 k 5 385 664 4 k 6 5 6456 4 k 5 3968 4 k 5 496 4 k 7 5 6 4 k 3 5 3 744 5 89 376 4 k 5 5 4 k 6 5 6 33 8 4 k 5 43 584 4 k 8 5 8 5 4 k 4 5 984 4 k 3 5 5 87 4 k 7 5 48 4 k 9 5 6 976 4 k 5
http://fuctios.wolfram.com G.Huvet (6) Itegral represetatios O the real axis Of the direct fuctio..7.. Ζs s s Ζs Ζs..7.. s t s t s t ; Res t t ; Res t..7.3. s Ζs s t s t..7.4. s t ; Res t t s t cschtt ; Res..7.5. s Ζs t s t sechtt ; Res s s..7.6. Ζs s s t s cschtt ; Res..7.7. Ζs s s t s csch tt ; Res..7.8. s Ζs t s sech tt ; Res s s..7.9. sis ta t Ζs t t s Π t s..7.. s Ζs s..7.. Ζs s t s frac t coss ta t t t s cosh Π t t ; Res
http://fuctios.wolfram.com..7.. Ζs s s s t t t t k s ; Res s k Ζs..7.. s s s B t t t s t..7.3. Ζs Πs ϑ s 3, Π t t s t ; Res k k s k B k ; Res Res s k..7.4. Πt logζs s t ; Res t t s For specific values..7.6. Ζ 3 Π..7.7. Ζ Π..7.8. Ζ Π..7.9. Ζ Π..7.. Ζ Π E x x ; E x ta Π x B x ta Π x E x cot Π x B x cot Π x x ; x ; x ; x ; Multiple itegral represetatios Ζs..7.. s logt Τ s..7.5. t Τ Ζ m m m tτ ; Res 3 t t t ; m m m k t k Product represetatios
http://fuctios.wolfram.com..8.. Ζs k p ; Res p s k primek k..8.. Ζs exp log Π s s k s s Ρ k ; ΖΡ k ImΡ k Ρ k Limit represetatios Ζs lim..9.. k k s s s ; Res s..9.. Ζs lim s k k k k s..9.3. m Ζ lim cot m m k k m ; Ζs Ζs lim z..9.4. lim s..9.5. k k z k ; z s k s k k s k F, k ; k ; This meas the classical series is Abel summable for all s. Differetial equatios Ordiary liear differetial equatios ad wroskias For the direct fuctio itself The zeta fuctio does ot satisfy ay algebraic differetial equatio (D. Hilbert, 9). Trasformatios Trasformatios ad argumet simplificatios Argumet ivolvig basic arithmetic operatios
http://fuctios.wolfram.com 3..6.. Π s s Ζ s Ζs s..6.. Ζ s Π cos Π s s s Ζs Ζ s..6.3. Π s s si Π s Ζs Idetities Fuctioal idetities For the fuctio itself..7.6. Ζs s s Π s si Π s..7.. Π s s Ζs Ζ s s Ζ s Ζs..7.3. s s k k s k j k j k j j Ζ j Krzysztof Maslaka: Hypergeometric-like Represetatio of the Zeta-Fuctio of Riema math-ph/57 () http://arxiv.org/abs/math-ph/57..7.5. Ζ k Ζ k k Ζ ; Icludig derivatives of the fuctio..7.. Ζ s k k exp s log Π Π k Π log Π exp s Π Π log Π log Π k k s Ζs ; s k
http://fuctios.wolfram.com 4..7.4. Ζ z k k z Π log Π k Π log Π z Πlog Π k Π log Π k z Ζz z k ; Differetiatio Low-order differetiatio Geeral case... Ζs logk ; Res s k k s...4. Ζs log k ; Res s k s k Derivatives at zero... Ζ log Π...7. Ζ Γ log Π Π...8. 4 Ζ 3 3 log Π Γ 3 Γ 3 Γ...9. Ζ3 log3 Π 8 Π log Π 3 log Π 3 Ζ 4 4 Π log Π Γ 6 log Π Γ 6 Γ 3 log Π Π 4 6 log Π Γ 6 log Π Γ Γ Γ 3 4 log Π Ζ3 log4 Π 4 3 log Π... 9 Π4 48 3 4 Ζ 5 96 8 3 Γ log Π Π 4 Π Γ 4 3 log Π Γ 3 log Π Γ Γ 3 4 Π 6 log Π Γ 3 Γ Ζ3 log 3 Π 4 6 log Π Γ 3 Γ Ζ3 log 3 Π Π log Π 48 log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 4 Ζ5 log Π Ζ3 log 5 Π 9 Π 4 log Π 7 4 log Π 9 5
http://fuctios.wolfram.com 5... Ζ 6 9 3 Π4 log Π Γ 5 log 4 Π Γ 5 8 4 8 Γ log Π Π 3 log 3 Π Γ 3 log Π Γ 3 5 log Π Γ 4 3 Γ 5 5 3 log Π Γ 6 Γ 8 Ζ3 4 log 3 Π Π log Π 5 8 Π log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 3 Π Γ log Π 4 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 9 Π 4 5 Π log Π Γ Γ 4 log 3 Π Ζ3 Γ 6 log Π Γ 4 log Π Γ 3 Γ 4 7 log Π Ζ5 Ζ3 log Π Γ Ζ3 6 Γ Ζ3 log 3 Π Ζ3 75 log6 Π 5 Π6 log Π 688... Ζ 7 384 344 5 6 Γ log Π Π 5 6 53 Π 4 6 log Π Γ 3 Γ Ζ3 log 3 Π 54 4 4 log Π Γ Γ Ζ3 log 3 Π Π log Π 56 3 Π Γ log Π 4 6 log Π Γ 6 log Π Γ Γ 3 8 log Π Ζ3 log 4 Π 9 Π 4 68 9 Π 4 Γ 4 Π 3 log Π Γ 3 log Π Γ Γ 3 48 5 log 4 Π 8 log Π Ζ3 Γ log Π Γ 3 5 log Π Γ 4 Γ 5 Γ log 3 Π Ζ3 336 Π log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 4 Ζ5 log Π Ζ3 log 5 Π 84 4 Π 6 log Π Γ 3 Γ Ζ3 log 3 Π 48 log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 4 Ζ5 log Π Ζ3 log 5 Π 9 Π 4 log Π 9 4 log 5 Π log Π Ζ3 4 Ζ5 Γ 4 log 3 Π Γ 3 5 log Π Γ 4 4 log Π Γ 5 7 Γ 6 7 Ζ7 54 log Π Ζ5 5 Γ log 4 Π 8 log Π Ζ3 8 log Π Ζ3 8 Γ 3 Ζ3 7 log 4 Π Ζ3 log 7 Π 75 Π 6 log Π 67 6 log Π 5 7
http://fuctios.wolfram.com 6...3. Ζ 8 75 96 Π6 log Π Γ 8 log 6 Π Γ 7 6 6 4 Γ 6 log Π 5 Π 84 log 5 Π Γ 4 log 4 Π Γ 3 4 log 3 Π Γ 4 84 log Π Γ 5 8 log Π Γ 6 4 Γ 7 8 5 6 log Π Γ 3 Γ 8 Ζ3 4 log 3 Π Π log Π 33 48 Π4 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 7 6 4 4 Π Γ 3 log Π 8 log Π Γ log Π Γ 4 Γ 3 4 log Π Ζ3 3 log 4 Π 9 Π 4 7 6 Π 3 log 4 Π 8 log Π Ζ3 Γ 6 log Π Γ 3 3 log Π Γ 4 6 Γ 5 7 44 log Π Ζ5 6 Γ log 3 Π Ζ3 4 Ζ3 4 log 3 Π Ζ3 log 6 Π 6 3 Π 6 log Π Γ 3 Γ 4 Ζ3 log 3 Π 4 log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 48 Ζ5 4 log Π Ζ3 log 5 Π 9 Π 4 log Π 96 596 Π 4 Γ log Π 68 Π log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 7 344 3 log 4 Π 8 log Π Ζ3 Γ 6 log Π Γ 3 3 log Π Γ 4 6 Γ 5 44 log Π Ζ5 6 Γ log 3 Π Ζ3 4 Ζ3 4 log 3 Π Ζ3 log 6 Π 75 Π 6 4 9 Π4 log Π Γ Γ Π 4 log 3 Π Ζ3 Γ 6 log Π Γ 4 log Π Γ 3 Γ 4 6 6 log 5 Π log Π Ζ3 4 Ζ5 Γ log 3 Π Γ 3 5 log Π Γ 4 6 log Π Γ 5 Γ 6 5 Γ log 4 Π 8 log Π Ζ3 4 Γ 3 Ζ3 88 log Π Ζ7 43 log Π Γ Ζ5 6 Γ Ζ5 344 Ζ3 Ζ5 67 log 3 Π Ζ5 Γ Ζ3 56 log Π Ζ3 log 3 Π Γ Ζ3 68 log Π Γ Ζ3 log Π Γ 3 Ζ3 8 Γ 4 Ζ3 56 log 5 Π Ζ3 83 log8 Π 4 7 Π8 log Π 3 4 7 8
http://fuctios.wolfram.com 7...4. Ζ 9 5 9 Ζ7 Γ 8 44 log Π Ζ5 Γ 8 log Π Ζ3 Γ 5 log 4 Π Ζ3 Γ 36 log 7 Π Γ 9 7 4 Γ 3 log Π Π 6 log 6 Π Γ 5 log 5 Π Γ 3 35 log 4 Π Γ 4 5 log 3 Π Γ 5 6 log Π Γ 6 36 log Π Γ 7 75 9 Γ 8 3 Π6 6 log Π Γ 3 Γ Ζ3 log 3 Π 6 4 6 log Π Γ 3 Γ Ζ3 5 log 3 Π 5 Π log Π 5 6 Π Γ log Π 4 3 log Π Γ 3 log Π Γ Γ 3 8 log Π Ζ3 log 4 Π 9 Π 4 399 8 Π4 log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 4 Ζ5 log Π Ζ3 log 5 Π 63 6 4 4 Π log Π Γ Γ Ζ3 log 3 Π 6 log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 7 Ζ5 6 log Π Ζ3 3 log 5 Π 9 Π 4 log Π 6 3 596 Π 4 Γ log Π 68 Π 6 log Π Γ 6 log Π Γ Γ 3 8 log Π Ζ3 log 4 Π 3 344 5 log 4 Π 8 log Π Ζ3 Γ 3 log Π Γ 3 5 log Π Γ 4 3 Γ 5 44 log Π Ζ5 3 Γ log 3 Π Ζ3 4 Ζ3 4 log 3 Π Ζ3 log 6 Π 75 Π 6 6 75 Π6 Γ 53 Π 4 3 log Π Γ 3 log Π Γ Γ 3 336 Π 3 5 log 4 Π 8 log Π Ζ3 Γ log Π Γ 3 5 log Π Γ 4 Γ 5 Γ log 3 Π Ζ3 9 7 log 6 Π 4 log 3 Π Ζ3 4 Ζ3 44 log Π Ζ5 Γ 35 log 4 Π Γ 3 35 log 3 Π Γ 4 log Π Γ 5 7 log Π Γ 6 Γ 7 Γ log 5 Π log Π Ζ3 4 Ζ5 8 log Π Γ 3 Ζ3 7 Γ 4 Ζ3 Π 4 log 5 Π log Π Ζ3 4 Ζ5 Γ 4 log 3 Π Γ 3 5 log Π Γ 4 4 log Π Γ 5 7 Γ 6 7 Ζ7 54 log Π Ζ5 5 Γ log 4 Π 8 log Π Ζ3 8 log Π Ζ3 8 Γ 3 Ζ3 7 log 4 Π Ζ3 log 7 Π 3 3 53 Π 4 6 log Π Γ 3 Γ Ζ3 log 3 Π 336 Π log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 4 Ζ5 log Π Ζ3 log 5 Π 9 4 log 5 Π log Π Ζ3 4 Ζ5 Γ 4 log 3 Π Γ 3 5 log Π Γ 4 4 log Π Γ 5 7 Γ 6 7 Ζ7 54 log Π Ζ5 5 Γ log 4 Π 8 log Π Ζ3 8 log Π Ζ3 8 Γ 3 Ζ3 7 log 4 Π Ζ3 log 7 Π 75 Π 6 log Π 6 Ζ9 96 log Π Ζ7 8 44 log Π Γ Ζ5 648 Γ 3 Ζ5 96 log Π Ζ3 Ζ5 5 log 4 Π Ζ5 Ζ3 3 54 Γ Ζ3 68 log 3 Π Ζ3 54 log 3 Π Γ Ζ3 54 log Π Γ 3 Ζ3 5 log Π Γ 4 Ζ3 54 Γ 5 Ζ3 84 log 6 Π Ζ3 log9 Π 83 Π 8 log Π 63 56 8 log Π 4 9
http://fuctios.wolfram.com 8...5. 83 Ζ 5 Π8 log Π Γ 45 log 8 Π Γ 5 8 8 4 Γ 84 log Π 7 Π 8 log 7 Π Γ 4 log 6 Π Γ 3 63 log 5 Π Γ 4 63 log 4 Π Γ 5 4 log 3 Π Γ 6 8 log Π Γ 7 45 log Π Γ 8 5 Γ 9 9 7 4 log Π Γ Γ 8 Ζ3 4 log 3 Π Π log Π 375 64 Π6 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 35 6 6 4 Π Γ 5 log Π 48 log Π Γ log Π Γ 4 Γ 3 4 log Π Ζ3 5 log 4 Π 9 Π 4 33 6 Π4 3 log 4 Π 8 log Π Ζ3 Γ 6 log Π Γ 3 3 log Π Γ 4 6 Γ 5 44 log Π Ζ5 6 Γ log 3 Π Ζ3 4 Ζ3 4 log 3 Π Ζ3 log 6 Π 5 Π 6 log Π Γ 3 Γ 8 Ζ3 4 log 3 Π log 3 Π Ζ3 Γ 3 log Π Γ 5 log Π Γ 3 5 Γ 4 96 Ζ5 8 log Π Ζ3 4 log 5 Π 9 Π 4 log Π 64 4 53 Π 4 Γ 3 log Π 56 Π log Π Γ log Π Γ 4 Γ 3 4 log Π Ζ3 3 log 4 Π 5 344 log 4 Π 8 log Π Ζ3 Γ log Π Γ 3 log Π Γ 4 Γ 5 44 log Π Ζ5 Γ log 3 Π Ζ3 4 Ζ3 4 log 3 Π Ζ3 log 6 Π 75 Π 6 8 Π 56 log 6 Π 4 log 3 Π Ζ3 4 Ζ3 44 log Π Ζ5 Γ 8 log 4 Π Γ 3 8 log 3 Π Γ 4 68 log Π Γ 5 56 log Π Γ 6 8 Γ 7 576 log Π Ζ7 68 Γ log 5 Π log Π Ζ3 4 Ζ5 688 Ζ3 Ζ5 344 log 3 Π Ζ5 log Π Ζ3 4 log Π Γ 3 Ζ3 56 Γ 4 Ζ3 log 5 Π Ζ3 log 8 Π 5 8 3 66 Π 4 6 log Π Γ 3 Γ 4 Ζ3 log 3 Π 68 Π log 3 Π Ζ3 Γ 3 log Π Γ log Π Γ 3 5 Γ 4 48 Ζ5 4 log Π Ζ3 log 5 Π 96 4 log 5 Π log Π Ζ3 4 Ζ5 Γ 4 log 3 Π Γ 3 5 log Π Γ 4 4 log Π Γ 5 7 Γ 6 44 Ζ7 8 log Π Ζ5 5 Γ log 4 Π 8 log Π Ζ3 56 log Π Ζ3 8 Γ 3 Ζ3 4 log 4 Π Ζ3 log 7 Π 75 Π 6 log Π 5 66 Π 6 Γ log Π 63 84 Π 4 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 5 6 88 Π 3 log 4 Π 8 log Π Ζ3 Γ 6 log Π Γ 3 3 log Π Γ 4 6 Γ 5 44 log Π Ζ5 6 Γ log 3 Π Ζ3 4 Ζ3 4 log 3 Π Ζ3 log 6 Π 5 56 log 6 Π 4 log 3 Π Ζ3 4 Ζ3 44 log Π Ζ5 Γ 8 log 4 Π Γ 3 8 log 3 Π Γ 4 68 log Π Γ 5 56 log Π Γ 6 8 Γ 7 576 log Π Ζ7 68 Γ log 5 Π log Π Ζ3 4 Ζ5 688 Ζ3 Ζ5 344 log 3 Π Ζ5 log Π Ζ3 4 log Π Γ 3 Ζ3 56 Γ 4 Ζ3 log 5 Π Ζ3 log 8 Π 83 Π 8 6 75 Π6 log Π Γ Γ 66 Π 4 4 log 3 Π Ζ3 Γ 6 log Π Γ 4 log Π Γ 3 Γ 4 Π 6 log 5 Π log Π Ζ3 4 Ζ5 Γ log 3 Π Γ 3 5 log Π Γ 4 6 log Π Γ 5 Γ 6 5 Γ log 4 Π 8 log Π Ζ3 4 Γ 3 Ζ3
http://fuctios.wolfram.com 9 4 5 6 3 48 8 log 7 Π 7 log 4 Π Ζ3 8 log Π Ζ3 54 log Π Ζ5 7 Ζ7 Γ 56 log 5 Π Γ 3 7 log 4 Π Γ 4 56 log 3 Π Γ 5 8 log Π Γ 6 8 log Π Γ 7 Γ 8 8 Γ log 6 Π 4 log 3 Π Ζ3 4 Ζ3 44 log Π Ζ5 344 Γ 3 Ζ5 log Π Γ 3 Ζ3 56 log Π Γ 4 Ζ3 Γ 5 Ζ3 6 log Π Ζ9 59 log Π Γ Ζ7 9 6 Γ Ζ7 86 4 Ζ3 Ζ7 43 log 3 Π Ζ7 36 88 Ζ5 6 48 log 3 Π Γ Ζ5 9 7 log Π Γ Ζ5 6 48 log Π Γ 3 Ζ5 5 Γ 4 Ζ5 96 Γ Ζ3 Ζ5 6 48 log Π Ζ3 Ζ5 34 log 5 Π Ζ5 log Π Ζ3 3 5 4 log Π Γ Ζ3 5 4 log Π Γ Ζ3 6 8 Γ 3 Ζ3 4 log 4 Π Ζ3 54 log 5 Π Γ Ζ3 6 log 4 Π Γ Ζ3 6 8 log 3 Π Γ 3 Ζ3 6 log Π Γ 4 Ζ3 54 log Π Γ 5 Ζ3 84 Γ 6 Ζ3 log 7 Π Ζ3 log Π 95 65 4 9 Π log Π 58...6. 9 3 4 5
http://fuctios.wolfram.com 3 4 5 3
http://fuctios.wolfram.com 3 8 7 log 7 Π 7 log 4 Π Ζ3 8 log Π Ζ3 54 log Π Ζ5 7 Ζ7 Γ 54 log 5 Π Γ 3 63 log 4 Π Γ 4 54 log 3 Π Γ 5 5 log Π Γ 6 7 log Π Γ 7 9 Γ 8 4 3 Ζ9 5 9 log Π Ζ7 5 Γ log 6 Π 4 log 3 Π Ζ3 4 Ζ3 44 log Π Ζ5 96 Γ 3 Ζ5 4 9 log Π Ζ3 Ζ5 34 log 4 Π Ζ5 4 Ζ3 3 336 log 3 Π Ζ3 8 log Π Γ 3 Ζ3 54 log Π Γ 4 Ζ3 8 Γ 5 Ζ3 68 log 6 Π Ζ3 log 9 Π 83 Π 8 log Π 84 4 Ζ 8 8 log Π Ζ9 45 6 log Π Γ Ζ7 475 Γ 3 Ζ7 95 4 log Π Ζ3 Ζ7 8 8 log 4 Π Ζ7 399 68 log Π Ζ5 33 64 log 3 Π Γ Ζ5 33 64 log Π Γ 3 Ζ5 66 3 log Π Γ 4 Ζ5 33 64 Γ 5 Ζ5 76 Ζ3 Ζ5 665 8 Γ Ζ3 Ζ5 76 log 3 Π Ζ3 Ζ5 5544 log 6 Π Ζ5 6 6 log Π Ζ3 3 77 log Π Γ Ζ3 84 8 log Π Γ 3 Ζ3 46 Γ 4 Ζ3 94 log 5 Π Ζ3 7 7 log 5 Π Γ Ζ3 46 log 4 Π Γ 3 Ζ3 46 log 3 Π Γ 4 Ζ3 7 7 log Π Γ 5 Ζ3 94 log Π Γ 6 Ζ3 3 Γ 7 Ζ3 65 log 8 Π Ζ3 log Π 95 65 Π log Π 48 99 log Π 5...7.
http://fuctios.wolfram.com 5 6
http://fuctios.wolfram.com 3 5 6 log Π
http://fuctios.wolfram.com 4 log Π Π Ζ
http://fuctios.wolfram.com 5 log Π 6 log Π 93 84 93 Π 3 77 36 Derivatives at other poits...3. Ζ logglaisher...8. Ζ 3 loga 8 log Π 6 Ψ5 3 Ψ 4 7...9. Ζ 5 loga 6 4 log Π Ψ7 6 Ψ 6 Ψ 5 37 5... Ζ 7 loga 6 4 log Π 54 Ψ9 5 Ψ 8 4 Ψ 7 7 Ψ 5... Ζ 9 3 loga 3 4 log Π 36 88 Ψ 8 44 Ψ 3 4 Ψ 9 54 Ψ 7 Ψ 5 79 33 64... Ζ Π Ζ ;...3. Ζ 4 6 log logπ Π Ζ...4. Ζ...5. Ζ 6 Π loga log Π...6. Ζ Π log Π Ψ Ζ Ζ ; Symbolic differetiatio Geeral case...5. Ζs log k ; Res s k k s
http://fuctios.wolfram.com 6 Derivatives at special poits...7. Imz Ζ Π a k Imz k Π k k ; z log Π Π a k s k s Ζs s...8. Ζ ; Fractioal itegro-differetiatio...6. Α Ζs sα s Α Α s Α s logk Α QΑ,, s logk ; Res k k s Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio... k s Ζss s ; Res k logk Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio... s Α Ζss sα Α Α, s logk sα k s logk Α Defiite itegratio...3. tσ Ζ t Σ t Σ Π Σ Ζ Σ t Σ ; Σ t A. Ivi : Some Idetities for the Riema Zeta Fuctio math.nt/359 (3) http://arxiv.org/abs/math.nt/359 3...4. 8 coslog t Ζ t t Π log t 4 A. Ivi : Some Idetities for the Riema Zeta Fuctio math.nt/359 (3) http://arxiv.org/abs/math.nt/359
http://fuctios.wolfram.com 7 Summatio Ifiite summatio..3.4. s k k Ζk s zk Ζs, z ; z k..3.5. Ψ z Ζk z k ; z z k..3.6. Ψ z k Ζk z k Ψ z ; z z k..3.7. Ζ k k k C k 4 k Π G.Huvet (6)..3.. k s k Ζk s k k..3.. s k Ζk s s Ζs k k k..3.3. s k Ζk s Ζ s, a k k k Ζs, a..3.8. k Ζk z k a Ψ a z z a a a z a logz a Ψa ; z a k k z k z k..3.9. k Ζk z j Ψ Ζ, Ζ z ; z j j j Ψ j Ζj, z Ζ, j, z Operatios Limit operatio
http://fuctios.wolfram.com 8 lim s..5.. Ζs s Represetatios through more geeral fuctios Through hypergeometric fuctios Ivolvig p F q..6.. Ζ F, a, a,, a ; a, a,, a ; ; a a a..6.. Ζ F, a, a,, a ; a, a,, a ; ; a a a Through Meijer G Classical cases for the direct fuctio itself..6.3., Ζ G,,,,,,, ;..6.4. Ζ G,,,,,,,, ; Through other fuctios..6.5. Ζs, s,..6.6. Ζs, s, s..6.7. Ζs s, s, Ζs S s..6.8...6.9. Ζs Li s ; Res..6.. Ζs Li s s..6.. Ζs Ζs,
http://fuctios.wolfram.com 9..6.. Ζs s Ζ s,..6.3. Ζs Ζs, k ; s k..6.4. q Ζs q s Ζ s, k ; q q k..6.5. Ζs H s Ζs, Represetatios through equivalet fuctios With related fuctios..7.. Ζz Z z exp ϑ z Zeros Sums over zeros Modulo the Riema hypothesis the followig the followig sums over the otrivial zeros of the Zeta fuctio hold:..3.. log4 Π ; lim T Ρ k ΖΡ k ReΡ k Ρ k..3.. Γ Π 8 ; lim T Ρ k Ρ k ΖΡ k ReΡ k..3.3. 3 3 Γ 3 3 Γ 7 Ζ3 8 ; 3 lim T 3 Ρ k Ρ k ΖΡ k ReΡ k..3.4. Γ 4 Γ 4 Γ 4 3 Γ 3 Π4 96 ; 4 lim T 4 Ρ k Ρ k ΖΡ k ReΡ k..3.5. 5 5 3 Γ 5 Γ Γ 5 5 Γ 5 6 6 Γ Γ 3 5 Γ 4 4 3 Ζ5 3 ; 5 lim T 5 Ρ k Ρ k ΖΡ k ReΡ k
http://fuctios.wolfram.com 3..3.6. 6 Γ 3 6 4 Γ Γ 3 Γ 6 3 Γ 4 3 3 Γ 9 Γ Γ 3 4 4 Γ Γ Γ 4 Γ 5 Π6 ; 96 6 lim T 6 Ρ k Ρ k ΖΡ k ReΡ k..3.7. 7 7 5 Γ 7 7 4 Γ 7 6 3 Γ Γ 3 7 4 36 Γ Γ Γ 4 7 4 Γ 6 Γ Γ 3 Γ Γ 4 7 7 Γ 3 Γ 4 Γ 3 Γ 6 Γ 3 Γ 5 7 7 Ζ7 8 ; 7 lim T 7 Ρ k Ρ k ΖΡ k ReΡ k..3.8. 8 Γ 4 4 Γ 3 Γ 3 Γ 8 6 Γ Γ 5 Γ Γ Γ 5 8 6 Γ 3 4 Γ 3 Γ 3 Γ 5 5 9 4 5 Γ 4 3 4 5 Γ Γ 3 Γ Γ 4 6 3 3 48 Γ Γ Γ 4 9 6 Γ 9 Γ Γ 3 Γ Γ 4 Γ 6 Γ 7 Γ 3 63 7 Π8 ; 6 8 8 lim T 8 Ρ k Ρ k ΖΡ k ReΡ k..3.9. Η j log4 Π j j j j j j Ζj ; lim T Ρ k Ρ k Η k s k ΖΡ k ReΡ k logs Ζs k k Theorems The Riema hypothesis o the zeros of the zeta-fuctio All otrivial zeros of Ζ(s) lie o the straight lie Res. The equivalet versio of the Riema hypothesis The Riema hypothesis is equivalet to Ρ A geeralizatio of this result due to Li, Bombieri, Lagarias is: Ρ Γ log4π ; ΖΡ ImΡ. s s l ss Πs s Ζs s Ρ Ρ A. Weil's "explicit formula"
http://fuctios.wolfram.com 3 Let Α be ay fuctio from C, s Αtexps tt. The the followig idetity holds (the Ρ -sum rus over all otrivial zeros of Ζz) Ρ Ρ logp j Αk logp j logp j p k j Αk logp j ΑlogΠ j k j k Αt t Αt Α et t t t The distributio of the zeros The sequece of zeros of Ζz alog the critical lie t is homogeeously distributed mod. Zeta fuctio regularizatio If k a kz is defied for Rez ad ca be aalytically cotiued to a domai cotaiig z, the lim z k a k a kz lim s a k s k Oe map with zeta fuctio The map h : defied by ht ΖΣ t, Ζ Σ t, Ζ Σ t,, Ζ Σ t with costat / < Σ < is dese i. Oe max-property If f z is ay ovaishig cotious aalytic fuctio i the disk z 4, the there exists a real tε such that max Ζz 3 tε f z ε. z4 4 Motgomery cojecture The two-poit correlatio fuctio R r for the zeros of Ζz o the critical lie is R r siπ r Π r. Keatig Saith cojecture For k, k the followig is cojectured: T lim Ζ k T t t Gk ak log T Gk Π k Here Gz is the Bares G fuctio ad ak p k j j k jk j p. Hughes Keatig O Coel cojecture For k, k 3 the followig is cojectured:
http://fuctios.wolfram.com 3 lim T s T Ζ s Gk kt ak log T Gk 3 Π kk Here the sum exteds over all zeros o the critical lie ad Gz is the Bares G fuctio ad ak p k j j k jk j p. GUE hypothesis A fixed set of otrivial zeros of Ζz behaves asymptotically like the eigevalues of a Gaussia uitary esemble. Asymptotical behaviour of zeros The umber Nt of zeros Γ of Ζ t t behaves asymptotically as Nt Π log t Ologt. Π The lattice-packig desity for ay covex symmetrical body The lattice-packig desity for ay covex symmetrical body i dimesios satisfies the iequality Ζ. The probability of a lattice poit to be visible The probability of a lattice poit from d beig visible from the origi (i.e., a d tuple of itegers is relatively prime) is Ζd. The scatterig matrix The scatterig matrix of the Laplace Beltrami operator i the modular domai has the form Ω Ω Ω Ζ Ω Ζ Ω. History L. Euler (737) P. G. L. Dirichlet P. L. Chebyshev B. Riema (859) J. Hadamard (893) H. vo Magoldt (894) Ch. J. de la Vallee Poussi (896) Applicatios iclude umber theory, Bose Eistei ad Fermi Dirac statistics, aalytic approximatio ad evaluatio of itegrals ad products, regularizatio techiques i quatum field theory, Scharhorst effect of quatum electrodyamics, Browia motio.
http://fuctios.wolfram.com 33 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a key to the otatios used here, see http://fuctios.wolfram.com/notatios/. Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: http://fuctios.wolfram.com/costats/e/ To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: http://fuctios.wolfram.com/.3.3.. This documet is curretly i a prelimiary form. If you have commets or suggestios, please email commets@fuctios.wolfram.com. -8, Wolfram Research, Ic.