THE SIZE OF THE LERCH ZETA-FUNCTION AT PLACES SYMMETRICWITHRESPECTTOTHELINE R(s) = 1/2

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1 KOREKTURY cmj tex THE SIZE OF THE LERCH ZETA-FUNCTION AT PLACES SYMMETRICWITHRESPECTTOTHELINE R(s = 1/2 Ramūnas Garunkštis, Andrius Grigutis, Vilnius (Received March 29, 2017 Abstract Let ζ(sbetheriemannzeta-function If t 68and σ >1/2,thenitis knownthattheinequality ζ(1 s > ζ(s isvalidexceptatthezerosof ζ(s Herewe investigate the Lerch zeta-function L(, α, s which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the criticalline However,forequalparameters =αitisstillpossibletoobtainacertain versionoftheinequality L(,,1 s > L(,, s Keywords: Lerch zeta-function; functional equation; zero distribution MSC2010:11M35 1 Introduction Let s = σ + itbeacomplexvariable In1965Spirain[20],Theorem2,proved thattheriemannhypothesisistrueifandonlyif (1 ζ(1 s > ζ(s, t 10, 1 2 < σ < 1 DixonandSchoenfeldin[4]showedthatif t > 68and σ > 1/2,then ζ(1 s > ζ(s exceptatthezerosof ζ(s Inequality(1wasstudiedbySaidakandZvengrowski in[18], Nazardonyavi and Yakubovich in[17], and Trudgian in[25] Also it was investigated for other zeta-functions Berndt in[3] generalized Spira s inequality for some functions of the class of general Dirichlet series, Spira in[21] proved itinthecaseoftheramanujan τ-dirichletseries,garunkštisandgrigutisin[6] considered the ana of inequality(1 for the Selberg zeta-functions We discuss a monotonicity of the modulus of the Riemann zeta-function Matiyasevich, Saidak, Supported by grant No MIP-049/2014 from the Research Council of Lithuania 1

2 andzvengrowskiin[16]notethat strictdecreaseofthemodulusofanycontinuouscomplexfunction falonganycurveinthecomplexplaneclearlyimpliesthat f canhavenozeroalongthatcurve Themonotonicityofthemodulusofacomplex function f isrelatedtothesignoftherealpartofthearithmicderivative Rf /f; see[16], Lemma 23 The well known Rieman ξ-function is defined as (2 ξ(s := 1 ( s 2 s(s 1π s/2 Γ ζ(s 2 Itsatisfies ξ(s = ξ(1 s,anditisanentirefunction,whosezeroscoincidewith nontrivialzerosof ζ(sitisknownthat R ξ (s > 0 when Rs > 1 ξ and the Riemann hypothesis is equivalent to R ξ ξ (s > 0 when Rs > 1 2 TheproofsofthelasttwoinequalitiescanbefoundbyHinkkanenin[13]orLagarias in[14]; see also Garunkštis[5] Sondow and Dumitrescu in[19] also investigated the relation between the monotonicity of ξ(s and the Riemann hypothesis Matiyasevich, Saidak, and Zvengrowski in[16] showed that R ζ (s < Rξ ζ ξ (s for t > 8and σ < 1Moreover,theyprovedthatthemodulusofthefunction ζ(s isdecreasingwithrespectto σintheregion σ 0, t 8;extendingthisregionto σ 1/2isequivalenttotheRiemannhypothesisThesimilarmodulusmonotonicity properties and the sign of the real part of the arithmic derivative of the Selberg zeta-functions are investigated by Grigutis and Šiaučiūnas in[12] See also Alzer[1] forthemonotonicityofthefunction F a (σ = (1 1/ζ(σ 1/(σ a,where a 1and σ > 1 InthispaperweconsidertheLerchzeta-functionWealwaysassumethat 0 <, α 1arefixedparametersTheLerchzeta-functionisgivenby 2 L(, α, s = m=0 e 2πim (m + α s, σ > 1

3 This function has an analytic continuation to the whole complex plane except for apossiblesimplepoleat s = 1 Let ζ(sand L(s, χdenotetheriemannzetafunction and the Dirichlet L-function, respectively We have that L(1, 1, s = ζ(s, L(1, 1/2, s = (2 s 1ζ(s, L(1/2, 1, s = (1 2 1 s ζ(s, and L(1/2, 1/2, s = 2 s L(s, χ, where χisanodddirichletcharacter mod4forthesefourcasescertainversionsof the Riemann hypothesis can be formulated For all the other cases, it is expected thattherealpartsofzerosofthelerchzeta-functionformadensesubsetofthe interval (1/2, 1 Thishasbeenprovedforany andtranscendental αandsome other cases; for more details see Laurinčikas and Garunkštis[15], Chapter 8 Only for these four above mentioned cases the Euler product and a symmetry(for most zeros with respect to the critical line is expected Nextwediscuszerofreeregionsanddefinetrivialandnontrivialzerosof L(, α, s For 1/2and 1let (3 l: σ = πt 1 bealineinthecomplexplane CIf = 1/2or = 1,thenlet lbeareallineofthe complexplanedenoteby h(s, lthedistancefrompoint sto ldefinefor ε > 0, + 1 L ε ( = { s C: h(s, l < ε } Let 0 < < 1and 1/2;then L(, α, s 0if σ < 1and s L 4/π ( Moreover,thereexistsaconstant δ 1 1suchthat L(, α, shasexactlyonezero with real part between 2π(α + k (4 σ k := 1 π + 1 π 2 1 and σ k+1 for σ k δ 1 (seegarunkštisandlaurinčikasin[7],garunkštisandsteudingin[9],lemma6,and[10] For = 1/2, 1,fromSpirain[22]and[7]wesee that L(, α, s 0if σ < 1and t 1Also,in[7]itisshownthat L(, α, s 0 for σ 1 + α Bythiswesaythatazeroof L(, α, sisnontrivialifitliesinthe strip 1 σ < 1 + αifazeroliesoutsidethestrip 1 σ < 1 + α,thenwecallit trivial Denoteby N(, α, Tthenumberofnontrivialzerosofthefunction L(, α, sin theregion 0 < t < TFor 0 <, α 1wehave[7] (5 N(, α, T = T 2π T 2πeα + O( T, T 3

4 Let = (, α = β + iγalwaysdenoteazeroof L(, α, s Forapositiveconstant Cwedefinethefollowingregionrelatedtothezerosof L(, α, s (6 R(, α, C = ( (,α β 1/2 ( (,α β 1 We prove the following theorem {s: s (, α < e Cγ/ γ, t 2} { s: s (1 (, α < 1 γ, t 2 } Theorem1Let 0 < 1Thenthereareconstants C > 0and t 0 = t 0 ( > 0 suchthatfor σ > 1/2 + e t, s / R(,, C,and t t 0, L(,, 1 s > L(,, s Theorem 1 implies the following corollary Corollary 2 Suppose the conditions of Theorem 1 are satisfied Then L(,, 1 s 0 WeexpectthatundertheconditionssimilartothosegiveninTheorem1the function f(σ = L(,, 1 σ+it isincreasing,similarlytothecaseoftheriemann zeta-function InthenextsectionTheorem1isproved 2 ProofofTheorem1 First we formulate several useful lemmas Lemma3 Wehave Γ(s = 2π s σ 1/2 e σ t arg s( ( O, t as s,uniformlyfor π + δ args π δ 4

5 Proof ThelemmafollowsfromStirling sformula(seetitchmarsh[23],section 442 ( Γ(s = s 1 s s + 1 ( π + O, s as s,uniformlyfor π + δ args π δ Lemma4 For 1/2 σ 1and t 2π + 1wehave Γ(s 2π s σ 1/2 e πt/2( 1 4σ3 σ 12t 2 (1 e πt Proof Toprovethelemmaitisequivalenttoshowthat We observe that (2π σ Γ(s e πt/2 ( s σ 1/2 ( 1 4σ3 σ 2π 12t 2 (1 e πt (2π σ Γ(s e πt/2 = (2π s 2Γ(scos πs e πs/2 2 2 cos(πs/2 Saidak and Zvengrowski in[18], Theorem 4, proved that (2π s 2Γ(scos πs 2 ( s σ 1/2 ( 1 4σ3 σ 2π 12t 2 if 1/2 σ 1and t 2π + 1Bytheinequality 1/(1 + x 1 x,where x > 1, weget e πs/2 1 1 = 2 cos(πs/2 1 + e πis 1 + e πt 1 e πt The lemma is proved Lemma5 If f(sisregular,and in {s: s s 0 r}with M > 1,then f(s 0 f(s f(s < e M f(s 0 s < e CM s 0 for s s 0 3r/8,where Cissomeconstantand runsthroughthezerosof f(s that s 0 r/2 5

6 Proof ThelemmafollowsimmediatelyfromtheproofofLemma αbytitchmarsh in[24], Paragraph 39 Lemma6 Forany σ 0 thereisaconstant A > 0suchthat (7 L(, α, σ + it = O(t A, t, uniformlyin σ σ 0 Proof For 0 < < 1,thisisTheorem14inChapter3ofLaurinčikasand Garunkštis[15] For = 1,thelemmafollowsfromtheHurwitzformula(Apostol[2], Theorem 126 and the Phragmén-Lindelöf theorem(titchmarsh[23], Paragraph 565 Lemma7 Let 0 <, α 1Let σ 0 Rand Rs σ 0 Let L(, α, s 0and d bethedistancefrom stothenearestzeroof L(, α, sthen 1 < exp(b( d + 1 t, L(, α, s where Bisapositiveconstant,whichdependson σ 0 andparameters αand Proof ThelemmaisprovedbyGarunkštisandTamošiūnasin[11],Proposition 2 We reproduce the proof here for completeness InLemma5wechoose f(s = L(, α, s, s 0 = 3 + it,andasufficientlylarge butfixedradius r InviewofLemma6wetake M = b T,where b = b(r Thefunction 1/L(, α, s 0 isboundedbytheformulaforthenumberofnontrivial zeros(5andinviewofthedistributionoftrivialzeros(seethediscussionnextto formula(4,thenumberofzerosinthedisc s s 0 < r/2islessthan c Is 0 This proves the lemma Lemma8 Let 0 <, α 1and ε > 0Let sbesuchthat min (,α s (1 (, α > 1 γ Thenforasufficientlylarge tandasufficientlylarge σ, (8 L(, α, 1 s > s (1 εσ 6

7 Proof Bythedistributionoftrivialzerosofthefunction L(, α, s(seethe discussionnexttoformula(3,inthehalf-plane σ 2thezerosofthefunction L(, α, 1 slieneartotheline k := { s: σ = πt } 1 if 1 2, 1, and k := {s: Is = 0} if = 1/2, 1 Firstweconsiderthecase 1/2 < < 1 Then (/(1 ispositiveandthe line kintersectsthearea {s: σ > 2, t > 0} Wewillinvestigateinequality(8for the following three subcases correspondingly: the value of s is(a above,(b below, and(cneartotheline k We start with the functional equation of the Lerch zeta-function(see for example Laurinčikas and Garunkštis[15], Chapter 2, or Garunkštis, Laurinčikas and Steuding[8],formula(1,whichwewriteintheform (9 L(, α, 1 s = (2π s Γ(s(e πis/2+2πiα L(α,, s + e πis/2 2πiα(1 {} L(1 α, 1 {}, s where {}denotesthefractionalpartofnumber Let ε > 0ByLemma3forany sufficiently large σ and sufficiently large t, (10 Γ(s (2π σ e πt/2 > ( s σ 1/2e ( ( σ O > s (1 εσ 2π t Subcase(aInviewofformula(9weseethatfor σ > 0andany t, (11 ( L(, α, 1 s = Γ(s (2π σ e πt/2 σ 1 + e πis 2πiα 1 {} (( ( s + + e πis 2πiα s + m 1 {} + m m=1 s We consider the function (12 g(s := 1 + e πis 2πiα( 1 {} (( + + m m=1 s s + e πis 2πiα ( 1 {} + m s 7

8 Thisisananalyticfunctionfor σ > 1If sliesontheline k,then e πis 2πiα( s = 1 1 {} If s kand t > 0,thenthereis δ > 0(independenton ssuchthatfor Iz δ + Is (ie zisabovetheline k, e πiz 2πiα( z ( πδ exp 1 {} Moreover,forsufficientlylarge σandany t > 0, (13 m=1 1 {} 1 3 (( ( s + e πis 2πiα s 1 + m 1 {} + m 3 Bythelasttwoinequalitiesweconcludethatthereare δ > 0, σ 0 > 0and t 0 > 0 such that (14 g(s 1 3 if t t 0 and (15 σ 0 σ π(t δ 1 {} Thus,if ssatisfiescondition(15,thenbyformulas(9and(10wegetforasufficiently large t and a sufficiently large σ, (16 L(, α, 1 s 1 3 Γ(s (2π σ e πt/2 σ > s (1 εσ Subcase (b Further we consider the half-plane beneath the line k By the functional equation(9 we see that (17 L(, α, 1 s > Γ(s (2π σ e πt/2 (1 {} σ ( 1 e πt( σ 1 {} ( e πt( + m σ ( 1 {} + m σ + 1 {} 1 {} m=1 Similarlyasabove,thereare δ 1 > 0,and t 1 > 0suchthat 8 1 eπt( σ 1 {} m=1 ( e πt( + m 1 {} σ ( 1 {} + m σ {} 3

9 if t t 1 and σ π(t + δ 1 1 {} From the inequalities above, inequality(8 follows for Subcase(b Subcase(c Let δand δ 1 beasinsubcases(aand(b,respectively Let s 1 = σ 1 + it 1 besuchthat (18 π(t 1 δ σ 1 π(t 1 + δ 1 1 {} 1 {} and min (,α s 1 (1 (, α = d > 0 WewillapplyLemma5toprovethatthereisaconstant C 1 = C 1 (, α, δ, δ 1 > 0 such that (19 g(s 1 > exp(c 1 ( d 1 Firstweshowthat g(sisboundedinanyfixedwidthneighborhoodoftheline k Moreprecisely,if δ 2 > 0and (20 σ π(t + δ 2, 1 {} then ( e πis 2πiα s ( exp 1 {} πδ 2 1 {} Thelastinequality,(12and(13givethatforasufficientlylarge tandasufficiently large σ satisfying inequality(20, (21 g(s 4 ( 3 + exp πδ 2 Let 1 {} (22 s 0 = σ 1 + i σ 1 π 1 {} + iδ, ie s 0 hasthesamerealpartas s 1,and s 0 liesontheline k + iδ := { s: σ = π(t δ 1 {} } 9

10 Thenby(14weseethat (23 g(s 0 > 1 3 The functional equation(9 and the definition(12 of the function g(s imply that for σ > 2,thesetofzerosof g(sisequaltotheset (24 {1 (, α: (, αisatrivialzeroof L(, α, s} WeapplyLemma5with f(s = g(s, s 0 definedby(22,and r = 8(δ + δ 1 /3 By inequalities(21 and(23 we choose ( π8(δ + δ1 /3 M = exp 1 Inviewofthedistributionoftrivialzerosof L(, α, s(seethediscussionnextto formula(4,andformula(24wehavethatthenumberofzerosof g(sinthedisc s s 0 < r/2islessthanorequelto C 2 = C 2 (, δ, δ 1 ThenLemma5givesthe desiredinequality(19 Let 1 (, αbethezeroof g(snearestto s 1 Inview ofthedistributionoftrivialzerosof L(, α, s,thereisaconstant C 3 = C 3 (, δ, δ 1 suchthat γ t 1 + C 3 Thus,choosing d 1/γweobtainfrominequality(19that forasufficientlylarge t 1, g(s 1 > exp(c 1 ( (t 1 + C 3 Thenbound(10togetherwith(11givesforlarge σ 1 andlarge t 1, L(, α, 1 s 1 > s 1 (1 εσ Bythis,(16,andSubcase(bweproveLemma8for 1/2 < < 1 If 0 < 1/2or = 1,then swithapositiverealandapositiveimaginarypart liesabovetheline k InthiscaseLemma8followsbyreasoningsimilartothatin Subcase(a The lemma is proved NowwewillproveTheorem1 Itisaconsequenceofthefollowingmoregeneral proposition Proposition9 Let ε > 0and 0 <, α 1 Thenthereareconstants C > 0 and t 0 = t 0 (, α, ε > 0suchthatfor σ > 1/2 + e t, s / R(α,, C,and t t 0, L(, α, 1 s > L(α,, s 10

11 Proof TheDirichletseriesof L(α,, sconvergesabsolutelyfor σ > 1,thus L(α,, sisboundedfor σ 2TheninviewofLemma8weconcludethatthereis σ 0 suchthatfor σ σ 0 andasufficientlylarge t,proposition9istrue Let 1/2 < σ σ 0 Inviewofthefunctionalequation(9andLemma4weobtain L(, α, 1 s = (2π σ Γ(s e πt/2 1 + e πi(s 2α(1+[] (25 L(α,, s ( t σ 1/2 ( 1 4σ3 σ 2π 12t 2 (1 e πt ( 1 e πt L(1 α, 1 {}, s L(α,, s L(1 α, 1 {}, s L(α,, s Let dbethedistancefrom stothenearestzeroof L(α,, snotethatthisnearest zeroisanontrivialzeroof L(α,, sif tislarge ByLemma7,thereisapositive constant B such that (26 1 < exp(b( d + 1 t L(α,, s Moreover,byLemma6,weseethat L(1 α, 1 {}, s = O(t A Thisandformulas(25,(26showthatthereisapositiveconstant Dsuchthat (27 L(, α, 1 s > L(α,, s ( t σ 1/2 ( 1 4σ3 σ 2π 12t 2 (1 e πt (1 De πt+b( d +1 t+a t Next we will prove that, under conditions of the proposition, Itiseasytoseethatfor 1/2 < σ σ 0, L(, α, 1 s > 0 L(α,, s ( (28 1 4σ3 σ ( σ 1/2 12t 2 = O t 2, t and (29 (1 e πt = O(e πt, t Thecondition s R(α, impliesthat d Ct/ t,where Cwillbechosen later Therefore, (30 (1 De πt+b( d +1 t+a t = O(e ( π+(bc+1t, t 11

12 Wechoose 0 < C < (π 3/B Thenbythecondition σ 1/2 > e t andby formulas(27 (30 we have L(, α, 1 s L(α,, s ( σ 1 ( t ( 1 2 2π + O t 2 + O(e ( π+(bc+1t > e t + O(e 2t > 0, t ThisprovesProposition9for 1/2 < σ σ 0 andfinishestheproof References [1] H Alzer: Monotonicity properties of the Riemann zeta function Mediterr J Math 9 (2012, [2] T M Apostol: Introduction to Analytic Number Theory Undergraduate Texts in Mathematics, Springer, New York, 1976 [3] BCBerndt:OnthezerosofaclassofDirichletseriesIIllJMath14(1970, [4] R D Dixon, L Schoenfeld: The size of the Riemann zeta-function at places symmetric withrespecttothepoint 1 2 DukeMathJ33(1966, [5] R Garunkštis: On a positivity property of the Riemann ξ-function Lith Math J 42 (2002, ; and Liet Mat Rink 42(2002, [6] R Garunkštis, A Grigutis: The size of the Selberg zeta-function at places symmetric withrespecttothelinere(s=1/2resultmath70(2016, [7] R Garunkštis, A Laurinčikas: On zeros of the Lerch zeta-function Number Theory and ItsApplicationsProcoftheConfHeldattheRIMS,Kyoto,1997(SKanemitsuet al, eds Dev Math 2, Kluwer Academic Publishers, Dordrecht, 1999, pp [8] R Garunkštis, A Laurinčikas, J Steuding: On the mean square of Lerch zeta-functions Arch Math 80(2003, [9] R Garunkštis, J Steuding: On the zero distributions of Lerch zeta-functions Analysis, München 22(2002, 1 12 [10] R Garunkštis, J Steuding: Do Lerch zeta-functions satisfy the Lindelöf hypothesis? Analytic and Probabilistic Methods in Number Theory Proc of the Third International Conf in Honour of J Kubilius, Palanga, 2001(A Dubickas, et al, eds TEV, Vilnius, 2002, pp [11] R Garunkštis, R Tamošiūnas: Symmetry of zeros of Lerch zeta-function for equal parameters Lith Math J 57(2017, [12] AGrigutis,DŠiaučiūnas:OnthemodulusoftheSelbergzeta-functionsinthecritical strip Math Model Anal 20(2015, [13] A Hinkkanen: On functions of bounded type Complex Variables, Theory Appl 34 (1997, [14] J Lagarias: On a positivity property of the Riemann ξ-function Acta Arith 89(1999, ; correction ibid 116(2005, [15] A Laurinčikas, R Garunkštis: The Lerch Zeta-Function Kluwer Academic Publishers, Dordrecht, 2002 [16] Y Matiyasevich, F Saidak, P Zvengrowski: Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions Acta Arith 166(2014, [17] S Nazardonyavi, S Yakubovich: Another proof of Spira s inequality and its application to the Riemann hypothesis J Math Inequal 7(2013,

13 [18] F Saidak, P Zvengrowski: On the modulus of the Riemann zeta function in the critical strip Math Slovaca 53(2003, [19] J Sondow, C Dumitrescu: A monotonicity property of Riemann s xi function and a reformulation of the Riemann hypothesis Period Math Hung 60(2010, [20] R Spira: An inequality for the Riemann zeta function Duke Math J 32(1965, [21] R Spira: Calculation of the Ramanujan τ-dirichlet series Math Comput 27(1973, [22] R Spira: Zeros of Hurwitz zeta functions Math Comput 30(1976, [23] E C Titchmarsh: The Theory of Functions Oxford University Press, Oxford, 1939 [24] E C Titchmarsh: The Theory of the Riemann Zeta-Function Oxford Science Publications, Oxford University Press, New York, 1986 [25] TSTrudgian:AshortextensionoftwoofSpira sresultsjmathinequal9(2015, Authors address: Ramūnas Garunkštis, Andrius Grigutis, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT Vilnius, Lithuania, ramunasgarunkstis@mifvult, andriusgrigutis@mifvult 13