Zeta. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

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1 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 ; Ζ Π B ; Ζ Π k k k r r r k r ; Ζ Π E ;

2 Values at fixed poits Ζ Ζ Ζ Ζ Ζ Ζ Ζ Ζ Ζ Ζ Ζ Ζ Ζ Π Ζ3 7Π3 8 k k 3 Π k k 5 k 6 k 3 k Ζ3 k k 5 k

3 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 Ζ3 3 Π 6 loga log Π Ψ Ζ F 7,,,,,, Brychkov Yu.A. (6) Ζ4 Π Ζ5 Π Ζ5 Π4 Ζ6 Π , 7 35 k k 5 Π k 35 k k 5 Π k log Π Ψ6 6 Ψ 5 Ψ Ζ7 9Π k k 7 Π k Ζ7 Π6 Ζ ; 4 3, 5 3, 3, 3,, 945 log Π 84 7 Ψ8 36 Ψ 7 6 Ψ 6 Ψ Π Ζ9 5Π Π k k 9 Π k 495 k k 9 Π k 8 6, Ζ9 475 log Π Ψ 5 Ψ 9 5 Ψ 8 4 Ψ 6 Ψ ; 7

4 4 Ζ Π Ζ 453Π k k Π k Π Ζ Ζ3 89Π Π4 Ζ k k 3 Π k 855 k k 3 Π k 3 687Π5 Ζ k k 5 Π k Π 6 Ζ Π 7 Ζ Π 8 Ζ k k 7 Π k Π 9 Ζ k k 9 Π k Π Ζ Π Ζ Π Ζ k k Π k 3 85 k k 7 Π k k k Π k

5 Π 4 Ζ Π 6 Ζ Π 8 Ζ Π 3 Ζ Π 3 Ζ Π 34 Ζ Π 36 Ζ Π 38 Ζ Π 4 Ζ Π 4 Ζ Π 44 Ζ Π 46 Ζ Π 48 Ζ

6 Π 5 Ζ Values at ifiities Ζ Geeral characteristics Domai ad aalyticity Ζs is a aalytical fuctio of s, which is defied over the whole complex s-plae 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 ig s Ζs,,, res s Ζs Brach poits The fuctio Ζs does ot have brach poits s Ζs Brach cuts The fuctio Ζs does ot have brach cuts.

7 s Ζs Series represetatios Geeralized power series Expasios at s s ; s Ζs Ζs Ζ s s s Ζ s s s ; s s s Ζs Ζs Ζ s s s Ζ s s s Os s 3 ; s 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 Ζs log Π s log logπ..6.. Ζs Os Γ Γ Π 4 s ; s Π 4 s Os 3 Expasios at s Ζs s Γ s Γ s ; s Ζs s Γ s Γ s Os Ζs k Γ s k s k k k

8 Ζs Os s 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 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 Ζs 4 Π s Ζ log4 Π Ψ Ζ Ζ O s 3 ; s Ζs Ζ Π s Os ; s Expoetial Fourier series 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 Ζs Ζs ; s This meas it caot be represeted through other fuctios. Other series represetatios Ζs ; Res s k k

9 Ζs k ; Res s s k k s Ζs s k ; Res s k..6.. Ζs s k k k k s Ζs 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 () Krzysztof Maslaka: {}[[,]] Acta Cosmologica XIII-, {}[[,]] (997) For specialized values m Ζ m m k k ; m m k m k Ζ k 4 k 4 k 4 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k 5

10 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 s Ζs s t s t s t ; Res t t s t cschtt ; Res s Ζs t s t sechtt ; Res s s Ζs s s t s cschtt ; Res Ζs s s t s csch tt ; Res s Ζs t s sech tt ; Res s s 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

11 Ζs s s s t t t t k s ; Res s k Ζs..7.. s s s B t t t s t Ζs Πs ϑ s 3, Π t t s t ; Res k k s k B k ; Res Res s k Πt logζs s t ; Res t t s For specific values Ζ 3 Π Ζ Π Ζ Π 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 t Τ Ζ m m m tτ ; Res 3 t t t ; m m m k t k Product represetatios

12 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 m Ζ lim cot m m k k m ; Ζs Ζs lim z lim s 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

13 Π s s Ζ s Ζs s..6.. Ζ s Π cos Π s s s Ζs Ζ s Π s s si Π s Ζs Idetities Fuctioal idetities For the fuctio itself Ζs s s Π s si Π s..7.. Π s s Ζs Ζ s s Ζ s Ζs 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 () 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

14 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 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 Π Ζ 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

15 Ζ Π4 log Π Γ 5 log 4 Π Γ Γ log Π Π 3 log 3 Π Γ 3 log Π Γ 3 5 log Π Γ 4 3 Γ 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 Ζ Γ log Π Π Π 4 6 log Π Γ 3 Γ Ζ3 log 3 Π 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 Π 4 Γ 4 Π 3 log Π Γ 3 log Π Γ Γ 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

16 Ζ Π6 log Π Γ 8 log 6 Π Γ 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 log Π Γ 3 Γ 8 Ζ3 4 log 3 Π Π log Π Π4 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 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 Π 3 log 4 Π 8 log Π Ζ3 Γ 6 log Π Γ 3 3 log Π Γ 4 6 Γ 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 Π 4 Γ log Π 68 Π log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 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 Π4 log Π Γ Γ Π 4 log 3 Π Ζ3 Γ 6 log Π Γ 4 log Π Γ 3 Γ 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

17 Ζ Ζ7 Γ 8 44 log Π Ζ5 Γ 8 log Π Ζ3 Γ 5 log 4 Π Ζ3 Γ 36 log 7 Π Γ 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 Γ 8 3 Π6 6 log Π Γ 3 Γ Ζ3 log 3 Π 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 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 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 Π 4 Γ log Π 68 Π 6 log Π Γ 6 log Π Γ Γ 3 8 log Π Ζ3 log 4 Π 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 Π6 Γ 53 Π 4 3 log Π Γ 3 log Π Γ Γ 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 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 Ζ log Π Γ Ζ5 648 Γ 3 Ζ5 96 log Π Ζ3 Ζ5 5 log 4 Π Ζ5 Ζ 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 log Π 4 9

18 Ζ 5 Π8 log Π Γ 45 log 8 Π Γ 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 log Π Γ Γ 8 Ζ3 4 log 3 Π Π log Π Π6 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 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 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 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 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 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 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 Π 4 log Π Γ log Π Γ 4 Γ 3 8 log Π Ζ3 log 4 Π 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 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 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

19 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 Ζ Ζ3 Ζ7 43 log 3 Π Ζ Ζ log 3 Π Γ Ζ5 9 7 log Π Γ Ζ log Π Γ 3 Ζ5 5 Γ 4 Ζ5 96 Γ Ζ3 Ζ log Π Ζ3 Ζ5 34 log 5 Π Ζ5 log Π Ζ 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 Π log Π

20

21 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 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 Ζ 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 Ζ log Π Γ Ζ7 475 Γ 3 Ζ log Π Ζ3 Ζ7 8 8 log 4 Π Ζ log Π Ζ log 3 Π Γ Ζ log Π Γ 3 Ζ log Π Γ 4 Ζ Γ 5 Ζ5 76 Ζ3 Ζ Γ Ζ3 Ζ5 76 log 3 Π Ζ3 Ζ log 6 Π Ζ5 6 6 log Π Ζ log Π Γ Ζ 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 Π log Π log Π

22 5 6

23 log Π

24 4 log Π Π Ζ

25 5 log Π 6 log Π Π Derivatives at other poits...3. Ζ logglaisher...8. Ζ 3 loga 8 log Π 6 Ψ5 3 Ψ Ζ 5 loga 6 4 log Π Ψ7 6 Ψ 6 Ψ 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 Ψ 8 44 Ψ 3 4 Ψ 9 54 Ψ 7 Ψ 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

26 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) coslog t Ζ t t Π log t 4 A. Ivi : Some Idetities for the Riema Zeta Fuctio math.nt/359 (3)

27 7 Summatio Ifiite summatio s k k Ζk s zk Ζs, z ; z k Ψ z Ζk z k ; z z k Ψ z k Ζk z k Ψ z ; z z k 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 s k Ζk s Ζ s, a k k k Ζs, a 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 k Ζk z j Ψ Ζ, Ζ z ; z j j j Ψ j Ζj, z Ζ, j, z Operatios Limit operatio

28 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,,,,,,, ; Ζ G,,,,,,,, ; Through other fuctios Ζs, s, Ζs, s, s Ζs s, s, Ζs S s Ζs Li s ; Res..6.. Ζs Li s s..6.. Ζs Ζs,

29 Ζs s Ζ s, Ζs Ζs, k ; s k q Ζs q s Ζ s, k ; q q k 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 Γ 3 3 Γ 7 Ζ3 8 ; 3 lim T 3 Ρ k Ρ k ΖΡ k ReΡ k Γ 4 Γ 4 Γ 4 3 Γ 3 Π4 96 ; 4 lim T 4 Ρ k Ρ k ΖΡ k ReΡ k Γ 5 Γ Γ 5 5 Γ Γ Γ 3 5 Γ Ζ5 3 ; 5 lim T 5 Ρ k Ρ k ΖΡ k ReΡ k

30 Γ Γ Γ 3 Γ 6 3 Γ Γ 9 Γ Γ Γ Γ Γ 4 Γ 5 Π6 ; 96 6 lim T 6 Ρ k Ρ k ΖΡ k ReΡ k Γ Γ Γ Γ Γ Γ Γ Γ 6 Γ Γ 3 Γ Γ Γ 3 Γ 4 Γ 3 Γ 6 Γ 3 Γ Ζ7 8 ; 7 lim T 7 Ρ k Ρ k ΖΡ k ReΡ k Γ 4 4 Γ 3 Γ 3 Γ 8 6 Γ Γ 5 Γ Γ Γ Γ 3 4 Γ 3 Γ 3 Γ Γ Γ Γ 3 Γ Γ Γ Γ Γ Γ 9 Γ Γ 3 Γ Γ 4 Γ 6 Γ 7 Γ Π8 ; lim T 8 Ρ k Ρ k ΖΡ k ReΡ k 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"

31 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:

32 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.

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