RAMŪNAS GARUNKŠTIS AND JUSTAS KALPOKAS

Transcription

1 SUM OF HE PERIODIC ZEA-FUNCION OVER HE NONRIVIAL ZEROS OF HE RIEMANN ZEA-FUNCION RAMŪNAS GARUNKŠIS AND JUSAS KALPOKAS Abstract We cosider the asymptotic of the sum of vaues of the periodic zeta-fuctio Ls, λ over otrivia zeros of the Riema zeta-fuctio ζs J Steudig for ratioa λ obtaied the mai term Assumig the Geeraized Riema Hypothesis we cacuate the ext term Itroductio Let s = σ +it deote a compex variabe ad defie ez = e πiz I this paper aways teds to pus ifiity ad p deotes a prime umber Let 0 < α ad λ R he Hurwitz zeta-fuctio is give by ad periodic zeta-fuctio by ζs, α = Ls, λ = + α s σ > eλ s σ > For α = λ = we get the Riema zeta-fuctio ζs = ζs, = Ls, he Riema hypothesis RH asserts that a otrivia o-rea zeros of ζs ie o the critica ie σ = / We wi eed the geeraized Riema hypothesis GRH Cosider the Dirichet L-fuctio χ Lχ, s = σ >, s where χ is a Dirichet character GRH states that i the critica strip 0 < σ the fuctio Lχ, s has zeros oy o the critica ie Let ρ = β + iγ deote the otrivia zero of the Riema zeta-fuctio Assumig RH, Fujii [] showed ζρ, α = π Λ α + L, α + O 9 0 og Steudig [6] proved the above formua ucoditioay with the error term O cog /3, where c is a absoute positive costat Assumig RH, Garuštis ad Steudig [5] improved Fujii s error term to O ε ad, assumig GRH, this error term ca be repaced by O +ε if α is ratioa For α = / we have ζs, / = s ζs hus, o RH, orea zeros of ζs, / ie o ies σ = 0 ad σ = / I [5] the behavior of sum was ivestigated whe α = α teds to / ad his ispired a Date: November, 007 AMS 99 subject cassificatio: M35, M6 Key words ad phrases: periodic zeta-fuctio, Riema zeta-fuctio, sum over zeros

2 R GARUNKŠIS AND J KALPOKAS cassificatio of otrivia zeros of the Riema zeta-fuctio depedety o the behavior of zeros of the Hurwitz zeta-fuctio For more detais see [5] I formua we see the periodic zeta-fuctio his fuctio is reated to the Hurwitz zeta-fuctio by the fuctioa equatio Aposto [], heorem 6 ζ s, α = Γs π s e s Ls, α + e s Ls, α Let ad be fixed atura umbers From Steudig [7] it foows that, for 0 < <,, =, L ρ, = e µ ϕ π og π + O he aim of this paper is to cacuate the term i the above formua For 0 < α we defie the geeraized Euer costat γα by = og + γα + O + α 0 We see that γ is the usua Euer costat For Dirichet character χ mod the Gauss sum is defied by a τχ = χae a= By χ 0 we deote the pricipa character he otatio p v meas that p v but p v+ heorem Assume GRH Let ad be fixed atura umbers with 0 < < ad, = he L s, = e µ ϕ π og π + C π + O 9/0 og, where C = e ϕ + µ + γ γ ϕ χ mod χ χ 0 χτχ + p L, χ L og p og p p p v p ϕ p v ϕ p v j= e p j p j I view of Ls, / = e / s ζs the costat C shoud be equa to zero his is ideed, because the sum over characters is empty ad γ = og + γ o prove the theorem we use Fujii s [] method We aso mae use of the sum Λ 3 = γ o cacuate this sum, et us cosider the Dirichet series Λ s

3 SUM OF HE PERIODIC ZEA-FUNCION OVER HE NONRIVIAL ZEROS 3 By the prime umber theorem this series coverges i the haf-pae σ For σ >, Λ = ζ s ζs s ζ Now we tae a imit s Lemmas We wi use the approximate fuctioa equatio for the periodic zeta-fuctio Lemma If 0 < λ, whe L + i, λ = π eλ +i + i πi i+ e π 0 π + λ + O i Proof his emma is the particuar case of the more geera approximate fuctioa equatio for the Lerch zeta-fuctio Lλ, α, s, which ca be foud i Garuštis, Lauričias, ad Steudig [] Note that Ls, λ = eλlλ,, s I the proof of heorem the mai too is the foowig two emas of Fujii [] Lemma For x > ad > 0 we have x γi = π + O x where Mx, = + O Λx x + Mx, + O x ogx x x <<x x og og og Λ mi, og x + O x og og og ogx, og og { x i og/π + O + πi og x og x og x, if og x, og O og mi og, og, if og x og x og Lemma 3 Suppose that 0 < πα Y <, ad > b πα 0 he we have for ay positive b, bγ e π og bγ = e πi α Λ b e α b πeα b Y <γ Y b πα b b πα b b + O 5 πα b + O b Y + Y og πα πα πα + O og mi, α + og Y πα We ofte wi use

4 R GARUNKŠIS AND J KALPOKAS Lemma Assume GRH If 0 < < is ratioa fixed umber with, =, whe Proof We have Λe = µ ϕ + O og Λe = a=0 e a a mod Λ he the emma foows by Daveport [3], Chapter 0 a mod Λ = ϕ + O og, ad Aposto [], Exercise a=0 a,= e a = a= a,= e a = µ Usig Lemma we have 3 Proof of heorem L, + γi = π + e πi + e πi π γ π + O 3 og e +γi e γ π og π γ γ πe =S + S + S 3 + O 3 og +γi e γ π og γ + +γi πe erm S By formua Daveport [3], Chapter 5 = π og π π + Oog

5 SUM OF HE PERIODIC ZEA-FUNCION OVER HE NONRIVIAL ZEROS 5 ad Lemma we get S =e π γ + π =e π og π π e + π π γ iγ e { Λ π + πi og π π + O πi og og + og + O og + O Λ mi, og <m<, m m og + O + O / og og og og og og og og } O / og ogπ ogπ og og ogπ + Λ i og π πi og Usig iequaities Λ og ad og + x x/ for Fujii [], we obtai x, simiary as i 5 S =e π og π π π + π Λe + O og Λe π By Lemma ad partia summatio we see that π Λe = µ ϕ π + O 3 og I 5 aother sum shoud be cacuated more precisey Euer s theorem tes us that, for p v, 6 p ϕ p v mod p v

6 6 R GARUNKŠIS AND J KALPOKAS his ad the orthogoa reatio of Dirichet characters give that Λe π = π,= = ϕ a= + og p p v p ϕ Λe + p e a χa χ mod p v ϕ p v j= e p j p j og p π If χ χ 0 the, usig the prime umber theorem, we obtai π Λ If χ = χ 0, the by formua 3 π Λ χ 0 = j= ep j p j Λ χ χ =L, χ Λ L > χ π = L L, χ + O / og Λχ 0 > π Λχ 0 + π Λ = og p + p og π + γ + O / og p = = og π og p p γ + O / og p From above, i view of a= e a χa = χτχ, it foows S = e µ ϕ π og π + L 7 χτχ, χ e + γ + ϕ L + χ mod p χ χ 0 ϕ p og p v e p j p v p ϕ 8 p v p j π + O 3/ og j= og p µ p ϕ

7 SUM OF HE PERIODIC ZEA-FUNCION OVER HE NONRIVIAL ZEROS 7 erm S Usig Lemmas 3 ad we get 9 S =e πi = π 6π γ e Λe = µ ϕ π + O 9 0 γ π og γ πe + O O erm S 3 Agai we use Lemmas 3 ad S 3 =e πi π = π = µ ϕ = µ ϕ + e γ π og π γ Λme + m π+ π π π + γ πe + m + O 9 0 og + π O 9 0 og + O 9 0 og he i view of ote that i the defiitio of the geeraized Euer costat the summatio starts from zero we have 0 S 3 = µ ϕ π og π + γ + µ ϕ Now the theorem foows by, 7, 9, ad 0 π + O 9 0 og Refereces [] M Aposto, Itroductio to Aaytic Number heory, Spriger-Verag, New Yor, 976 [] A Fujii, Zeta Zeros, Hurwitz Zeta Fuctio ad L, χ, Proc Japa Acad, Ser A, , 39 [3] G Daveport, Mutipicative Number heory Marham, Chicago, 967 [] R Garuštis, A Lauričias, ad J Steudig, A approximate fuctioa equatio for the Lerch zeta-fuctio, Math Notes, 7 003, [5] R Garuštis ad J Steudig, O the Distributio of Zeros of the Hurwitz Zeta-fuctio, Math Comput, , [6] J Steudig, O the vaue-distributio of the Hurwitz zeta-fuctio at the otrivia zeros of the Riema zeta-fuctio, Abhdg, Math Sem Ui Hamburg, 7 00, 3 [7] J Steudig, Dirichet series associated to periodic Arithmetic Fuctios ad the zeros of Dirichet L-fuctios Aa Probab Methods Number heory, 00, 8 96

8 8 R GARUNKŠIS AND J KALPOKAS Ramūas Garuštis Departmet of Mathematics ad Iformatics Viius Uiversity Naugarduo 600 Viius Lithuaia ramuasgarustis@mafvut Justas Kapoas Departmet of Mathematics ad Iformatics Viius Uiversity Naugarduo 600 Viius Lithuaia justasapoas@mafvut