AN APPROXIMATE FUNCTIONAL EQUATION FOR THE LERCH ZETA-FUNCTION. R. Garunkštis, A.Laurinčikas, J. Steuding

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1 AN APPROXIMATE FUNCTIONA EQUATION FOR THE ERCH ZETA-FUNCTION R. Garunkšis, A.aurinčikas, J. Seuding Absra. We prove an approximae funional equaion for he erh zea-funion eλn nα wih fixed parameers < λ, α 1. s 1. Inroduion As usual, le s = σ i, ez = expπiz and denoe by [λ] he inegral and by {λ} he fraional par of a real number λ. For < α 1, λ R, he erh zea-funion is given by eλn n α s. 1 This series onverges absoluely for σ > 1. Obviously, we have {λ}, α, s, so in he sequel we may assume < λ 1. If < λ < 1, hen he series 1 onverges even for σ >. Moreover, one an prove he funional equaion s λ, α, 1 s = π s Γs e 4 αλ α, λ, s e s 4 α1 λ α, 1 λ, s, and λ, α, s urns ou o be an enire funion. The firs proof of his funional equaion was given by erh [3]. For λ = 1 he erh zea funion redues o he Hurwiz zea-funion ζs, α := 1, α, s, whih an be oninued analyially o he whole omplex plane exep for a simple pole a s = 1, and saisfies he ideniy π1 s ζ1 s, α = π s Γs sin osπnα n s π1 s os sinπnα n s, 3 whih is valid for σ > if < α < 1, resp. for σ > 1 if α = 1. This formula was firs proved by Hurwiz []. For α = λ = 1 he erh zea-funion beomes he well known Riemann zea-funion ζs := ζs, 1. We reall he funional equaion ζs = χsζ1 s, 4 χs := s π s 1 sin πs Γ1 s. Siegel found in he unpublished noes of Riemann he approximae funional equaion see [5] or [6] : le σ 1, m = [ /π] and N < A wih some suffiienly small onsan A. Then m 1 ζs = n s χs m 1 n 1 s AN N/6 1 m 1 e πis 1/ π s/ 1/ e i/ iπ/8 Γ1 s S N O Oe A, 5 1

2 S N := N 1 ν n/ n! i ν n ν!n ν! n n/ ν a n ψ n ν η π π m wih a n defined in 1 below, and ψa := osπa / a 1/8. 6 osπa For Hurwiz zea-funions an approximae funional equaion a σ = 1/ was obained by Rane [4]. Exending hese resuls we will prove an approximae funional equaion for erh zea-funions. Noe ha we an unify he formulas, 3 and 4 by s λ, α, 1 s = π s Γs e 4 αλ α, λ, s e s 4 α1 {λ} α, 1 {λ}, s. For < λ < 1, < σ σ 1 and πλx he erh zea-funion an be approximaed by a finie sum see [1]: eλn m α s O σ x σ. 8 m x For appliaions he las sum is ofen oo long. A beer approximaion is given by Theorem e < λ 1, < α 1 and σ 1. e 1, y = /π 1/, q = [y], m = [y α] and β = q m. Then m eλk π k α s π σ 1 i q πi i e 4 πi{λ}α e αn n λ 1 s σ e πifλ,α,σ, ψy q β {λ} α O σ, fλ, α, σ, = π log πe α {λ} αβ yβ {λ} α 1 q m {λ}β α. 7 Sine 1 {λ}, α, s i is easy o obain also an approximaion for λ, α, s for negaive. Furher, noe ha he error erm in he Theorem an be sharpened and wrien like in formula 5.. Proof of he Theorem Suppose ha σ > 1. Then in view of he well known formula e αmz z s 1 dz = Γs α m s we obain m eλk k s s eπi{λ}m e mαz Γs e z πi{λ} 1 zs 1 dz.

3 We ransform he inegral ino a loop inegral as in [4],.4, or [3]. This yields m eλk k s s eπi{λ}m s/ Γ1 s πi G e mαz e z πi{λ} 1 zs 1 dz, G sars and finishes a, runs around he origin in posiive direion and exludes all zeros of e z πi{λ} 1 oher han z = if λ = 1. By analyi oninuaion he formula above holds for σ <. We deform he onour G ino sraigh lines G 1, G, G 3, G 4, joining, η iη1 πi{λ}, η iη1 πi{λ}, η q 1πi πi{λ},, := 3/4 and η := πy. If y is an ineger, a small indenaion is made above he pole a z = iη πi{λ}. In he same way as in erh [3] we obain by he alulus of residues m eλk k s s e πi{λ}m s/ Γ1 s eπi{λ}m s/ Γ1 s πi q n= q n iff λ=1 G 1 Res z=πin{λ} G G 3 G 4 e mαz e z πi{λ} 1 zs 1 e mαz e z πi{λ} 1 zs 1 dz. Firs we onsider he inegrals. Denoe by A a posiive onsan no neessarily always he same. e z = u iv = e iφ, φ < π. Then z s 1 = σ 1 e φ. On G 4 we have φ 5 4π, > Aη, and e z πi{λ} 1 > A. Hene for every ε >. Sine G 4 η σ 1 e 5π/4 η argθ = e mαu du e mαη 5π/4 e ε 5π/4 θ dµ θ 1 µ > dµ 1 µ = θ 1 θ, for θ >, i follows ha arg/1 > A. Thus, arg/1 πλ/η > A for suffiienly large. So we have on G 3, for > =, Hene and e z πi{λ} 1 > A. Thus φ 1 π arg Sine e z πiλ 1 > Ae u on G 1, i follows ha For < θ < 1, 1 πλ/η > 1 π A. z s 1 e mαz η σ 1 e 1 πamη η σ 1 e 1 πa, G 3 η σ e 1 πa. z s 1 e mαz e z πiλ 1 1 η π{λ} ησ 1 exp arg m 1 αu. u arg θ < θ dµ 1 µ = θ 1 θ, 3 9

4 herefore So for suffiienly large arg arg Sine m 1 α y = /η, and d 1 η π{λ} arg u du u η we have Hene arg 1 η π{λ} u G 1 η σ 1 1 < A. 1 π{λ}/η < A. u η arg 1 πη 1 η π{λ} = u 1 η π{λ} 1 η >, π{λ} η = π arg 1 π{λ}/η > 1 π A, e 1 πa du η σ 1 e xu du η σ e 1 πa η σ 1 e πηx η σ e 1 πa. Finally, we onsider G. Tihmarsh in [6], 4.16 wries Ψz := exp s 1log 1 z iz 1 iz =: πη a n z n 1 and shows ha a = 1, a n n/[n/3] no neessarily uniformly in n. Moreover, he proves for ha for N 7/5, z N /5 1/3 /1, and for z /. On G we have Whene G N 1 r N z := Ψz a n z n, N/3 5e r N z z N N r N z exp 9 z 1 z s 1e mαz = e gz Ψ iη z iη i π, gz := η i z iη π 4π z iη m αz. e mαz e z πi{λ} 1 zs 1 dz = G G iη s 1 e gz N 1 n z iη a e z πi{λ} n dz 1 iπ 1/ iη s 1 e gz e z πi{λ} 1 r N 4 z iη i π dz =: G 1 G.

5 For z G we have z = iη πi{λ} µe πi/4, µ is real and µ η. Then e gz e µ 4π π{λ} µ, iη s 1 η σ 1 e 1 π. If e z πiλ 1 > A on G, we obain wih regard o 11 and 1 AN 1/3 G η σ 1 e 1 π e µ 4π N N/3 µ 5e π{λ}µ π N dµ η/ AN 1/3 e µ 4π π{λ}µ 7µ πi πi{λ}e 4 9π 1 µ dµ 13 η σ 1 e 1 π 1 6 N e A/6 η σ 1 e 1 π N 6 for suffiienly large and fixed N. If he ondiion e z πi{λ} 1 > A is no saisfied suppose for example, ha he onour goes oo near o he pole a z = πiq {λ}, hen ake i round an ar of he irle z πiq {λ} = π/. On his irle z = πiq {λ} 1 πeiθ and So he onribuion a his par is From his and 13 i follows ha e gz A, z iη r N i N/6. π η σ 1 e 1 π N/6. G η σ 1 e 1 π N 6. In he inegral G 1 we replae onour G by he infinie sraigh line of whih i is a par, G his he inegral G 1 hanges by η σ 1 e 1 π e µ N 1 4π π{λ} µ η n/[n/3] πiλ µe 1 4 πi π dµ e 1 πa. say. Afer Thus N 1 G 1 = iη s 1 a n i n π n/ G e gz e z πi{λ} 1 z iηn dz Oe 1 πa. The inegral may be expressed as exp gz πiq {λ} z πiq {λ} iη n e z dz, 1 is a line in he direion argz = π/4, passing beween and πi1 {λ}. This is n! imes he oeffiien of ξ n in I = exp gz πiq {λ} ξ z πiq {λ} iη dz e z 1 = exp iπq {λ} iηq {λ} 3 iη 4 iz η dz exp 4π z π q {λ} m α ξ e z 1. 5 π iξπq {λ} η

6 However, by.1 of Tihmarsh [6], e iz /4πaz e z dz = πe iπa /3/8 ψa, 1 whene, afer simplifiaion, in view of expiπm / πq / πqm = exp iπβ/ πq, πi I = πgψ y q m ξexp ξ exp πiξq m, G = expπi y 38 1 α {λ} αβ yβ {λ} α 1 q m {λ}q m α, q = q {λ} and m = m α. Therefore, he oeffiien of ξ n in I is πg l! µ µ!ν! 1 ψ l y q m πi µν q m ν. lµν=n Here y q m 1, whene ψ l y q m 1, l N, and q m 1, so ha he oeffiiens are 1. Now we se N = 3. Sine a n n/ for n = 1, we may onlude ha e m z e z πi{λ} 1 zs 1 dz = πgiη s 1 ψy q m O 1/. G Noe ha by Sirling s formula and e πi{λ}m s/ Γ1 s = π 1 σ i e π i iπ σ 1 πi πi{λ}m 1 O 1, e πi{λ}m s/ Γ1 s πgiη s 1 = πi σ π e πifλ,α,σ, 1 O 1. We omplee he proof by alulaing he sum of residues in 9 and obain m eλk k α s π σ 1 q i πi i e 4 πi{λ}α e αn eπisπα n λ 1 s σ π e ifλ,α,σ, ψy q β {λ} α O σ. q eαn n 1 {λ} 1 s This should be ompared wih he funional equaion of erh zea-funions 7. Herein we an esimae he erm wih he hird sum on he righ hand side by e π. Thus we have proved he Theorem. As a simple appliaion of he approximae funional equaion i follows ha λ, α, 1/ i 1/4, as urns o infiniy, as he approximaion 8 leads only o he upper bound 1/. Referenes 1. R. Garunkšis, A. aurinčikas, On he erh zea-funion, ih. Mah. J , A. Hurwiz, Einige Eigenshafen der Dirihleshen Funkionen Fs = D 1 n n, die bei der Besimmung der Klassenzahlen binärer quadraisher Formen aufreen, Zeishrif f. Mah. u. Physik 7 s 1887, M. erh, Noe sur la fonion Kz, x, s = ekπix z k s, Aa Mah , V.V. Rane, On he mean square value of Dirihle -series, J. ondon Mah. So., 1 198, C.. Siegel, Über Riemanns Nahlass zur analyishen Zahlenheorie, Quellen u. Sudien z. Geshihe der Mah. Asronomie und Physik, Ab. B: Sudien 193, E.C. Tihmarsh, The heory of he Riemann zea-funion, nd ed., revised by D.R. Heah-Brown, Oxford Universiy Press