Approximation of the Lerch zeta-function

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1 Approximaion of he Lerch zea-funcion Ramūna Garunkši Deparmen of Mahemaic and Informaic Vilniu Univeriy Naugarduko Vilniu Lihuania ramunagarunki@mafvul Abrac We conider uniform in parameer approximaion of he Lerch zeafuncion by Dirichle polynomial I allow u o obain uniform in parameer bound in he criical rip Le = σ + i be a complex variable For σ > he Lerch zea-funcion i given by he erie (m + α) wih parameer λ IR and 0 < α Thi funcion can be analyically coninued o he whole complex plane wih a poible excepion of a pole a he poin = Noe ha L(λ α ) i periodic in λ In [] Chaper 3 we ee ha for fixed 0 < λ < and for σ σ 0 > 0 πλx he following approximaion hold x (m + α) + O ( x σ) In hi paper we preen a uniform in λ and α approximaion of he above ype We alo calculae an explici conan in he error erm Le Θ(z) denoe ome complex number uch ha Θ(z) z Parially uppored by Gran from Lihuanian Foundaion of Sudie and Science

2 Theorem Suppoe ha / λ / λ 0 and le σ 0 x Then (( ) ) 4 (m + α) + Θ + 05 x σ We give he proof a he end of he paper By he eimae m (m + α) σ ( + α) σ + dx (x + α) σ ( + ( σ) α) σ if 0 σ e ( + α) if σ we obain he following: Corollary Suppoe ha / λ / λ 0 and le σ 0 Then ( + ( σ) α) σ L(λ α ) α σ + 7 λ σ + if 0 σ e ( + α) if σ To find an aympoic dependence on he parameer λ and α we need an approximae funcional equaion Theorem 3 Le 0 < λ 0 < α and 0 < σ Moreover le y = (/(π)) / q = [y] k = [y α] and β = q k () where k + ( π ( π + e(λm) (m + α) ( ) σ q +i πi i+ e 4 πiλα ) e( αm) e π+πiσ+πiα (m + λ) ( {λ}) ) σ e if(λασ) ψ(y q k {λ} α) + O( σ ) f(λ α ) = π πe (α {λ} ) αβ + y(β + {λ} α) (q + k) {λ}(β + α)

3 and ψ(a) = co(π(a / a /8)) co(πa) Formula () immediaely follow from formula () of Chaper 4 of [] Though no menioned in [] one can ee from he proof of () ha formula () hold uniformly in λ and α From Theorem 3 we derive he following: Corollary 4 Le 0 < λ α σ 0 and Then L(λ α ) α ( ( ) π ) σ +i πi i+ e 4 πiλα e π+πiσ+πiα λ ( {λ}) σ if 0 < σ ( σ) if σ uniformly in λ and α For negaive we can ue he formula L(λ α σ + i) = L( λ α σ i) To prove Theorem we need he following: Lemma 5 Le f(x) be a real-valued funcion on [a b] uch ha f (x) i coninuou and monoonic on [a b] and f (x) δ < Then b ( 4 δ e (f(n)) = e (f(x)) dx + Θ π( δ) + 6 ) δ + 3 π a<n b Here e(x) = e πix a Proof of hi lemma can be found for example in Ivić [3]; concerning he explici conan ee Lemma 9 of [] Proof of Theorem Le for u > 0 Then () S(λ u) = m u S(λ u) = S(λ [u]) = eπiλ([u]+) e πiλ = O ( λ ) Now ummaion by par how x (m + α) = S(λ x)(x + α) du S(λ u) (u + α) +

4 Thu we have ha for σ > and any poiive ineger N N S(λ N)(N + α) (m + α) du + S(λ u) N (u + α) + Since he laer inegral converge uniformly on compac ube of he half-plane σ > 0 he above expreion alo remain valid for σ > 0 In view of () N du S(λ u) (u + α) = O ( N σ σ λ ) + and herefore we obain he following approximaion of he funcion L(λ α ) for σ > 0: N S(λ N)(N + α) (m + α) (3) Now le u conider he um (m + α) = + O ( N σ σ λ ) πi(λm ( (m+α))/π) e (m + α) σ Le f(u) = λu ( (u + α))/π Then f (u) = λ /π(u + α) and f (u) λ < for u [x N] Conequenly by Lemma 5 and inegraion by par A(N) := (m + α) i = Hence by parial ummaion we find N x = Θ (m + α) = A(N) N N + α + σ x e πi(λu ( (u+α))/π) du + Θ(05) ( ) A(u) du (u + α) σ+ Now in view of (3) ending N o infiniy we obain (( ) 4 (m + α) + Θ + 05 The la formula i proved for σ > 0 By he coninuiy of he Lerch zea-funcion i alo remain valid for σ 0 x σ ) 4

5 Reference [] R Garunkši The effecive univeraliy heorem for he Riemann zea funcion in: Proceeding of he eion in analyic number heory and Diophanine equaion MPI-Bonn January - June 00 Ed by D R Heah-Brown B Z Moroz Bonner mahemaiche Schrifen 360 (003) pp [] A Laurinčika and R Garunkši The Lerch Zea-funcion Dordrech Kluwer Academic Publiher (00) [3] A Ivić The Riemann Zea-funcion New York John Wiley (985) Reziumė Sraipnyje nagrinėjame olygia paramer u ažvilgiu Lerch o dzea funkcijo aprokimacija Dirichle polinomai Tai leidžia gaui olygiu paramer u ažvilgiu i verčiu kriinėje juooje 5