SIMULTANEOUS NONVANISHING OF DIRICHLET L-FUNCTIONS AND TWISTS OF HECKE-MAASS L-FUNCTIONS. 1. Introduction

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1 SIULTANEOUS NONVANISHING OF DIRICHLET L-FUNCTIONS AND TWISTS OF HECKE-AASS L-FUNCTIONS SOUYA DAS AND RIZWANUR KHAN Abstract. W prov that givn a Hck-aass form f for SL, Z and a sufficintly larg prim, thr xists a primitiv Dirichlt charactr χ of conductor such that th L-valus L, f χ and L, χ do not vanish. W xpct th sam mthod to work for any larg intgr.. Introduction Undrstanding whn an L-function s cntral valu is non-zro is a problm of grat significanc and long history in analytic numbr thory. Undrstanding whn two or mor L-functions ar simultanously non-zro is also a popular topic and can hav important applications. For xampl, th problms considrd in [9] and [] hav a baring on th Landau-Sigl zro problm and th Birch-Swinnrton-Dyr conjctur rspctivly. Rcntly thr has bn som intrst in th simultanous non-vanishing of L-functions in th family of primitiv Dirichlt charactrs. Blomr and ilićvić [] provd that givn two fixd Hck-aass forms f and f which satisfy th Ramanujan-Ptrsson conjctur, and a sufficintly larg intgr r subjct to a tchnical condition that dos not prmit intgrs such as prims and th product of two prims of almost ual siz, thr xists a primitiv Dirichlt charactr χ of modulus r such that L/, f χ and L/, f χ ar not zro. This improvd th work of Hoffstin and L [6], who studid th sam kind of non-vanishing problm, but wr not abl to spcify th modulus of th charactr. W study a simplr problm: that of th simultanous non-vanishing L/, f χ and L/, χ for a fixd Hck-aass form f for SL, Z. Our papr is motivatd by a rcnt rsult of Liu [], who showd that for R larg nough, thr xists a Dirichlt charactr modulo r with R < r < R such that L/, f χ and L/, χ ar non-vanishing. Actually Liu considrs mor gnral automorphic forms, but th point is that th modulus of th dsird Dirichlt charactr is not spcifid by his work. Liu s rsult follows by showing that. r D χ mod r χ = L/, f χl/, χ = r D r + OR 5 8 +ɛ for som subst D of th intgrs btwn R and R of siz D R/log R, whr indicats that th sum is rstrictd to th primitiv charactrs. Liu nots that sinc th right hand sid is non-zro, at last on of th summands on th lft hand sid must b non-zro. Th main difficulty in Liu s analysis sms to b in daling with Gauss sums, which aris from approximat functional uations. W will prov th following rsult. 000 athmatics Subjct Classification. Primary F; Scondary F. Ky words and phrass. L-functions, Cntral valus, Simultanous nonvanishing.

2 SOUYA DAS AND RIZWANUR KHAN Thorm.. Lt f b a Hck-aass form for SL, Z. For prim valus of, w hav that. χ mod χ = L, f χl, χ = L, f + O 7 8 +θ+ɛ, whr ɛ > 0 is arbitrarily small. Th implid constant dpnds on f and ɛ. In th xponnt, θ rprsnts th bst bound towards th Ramanujan-Ptrsson conjctur for f, which can currntly b takn to b θ = 7/6. By considring such a man valu with a complx conjugat, w avoid som of th difficultis with Gauss sums and do not nd to avrag ovr th modulus as in Liu s work. Sinc L, f is non-zro s [8, Lmma 5.9], w gt Corollary.. Fix f a Hck-aass form for SL, Z. For vry larg nough prim, thr xists a primitiv Dirichlt charactr χ of conductor such that th cntral L-valus L, f χ and L, χ do not vanish. Thus in our work, th modulus of th dsird Dirichlt charactr is known. W hav workd with prim moduli to minimiz tchnical dtails. Howvr givn th powr saving rror trm in., w xpct th sam mthod would yild th non-vanishing rsult for any larg intgr. Blomr, Fouvry, Kowalski, ichl, and ilićvić [] hav provn, conditionally on th Ramanujan- Ptrsson conjctur for f, th imprssiv asymptotic.3 χ mod L, f χl, χ = L, f ζ + O δ, for prim and som δ > 0. Although th man valu w considr is mor modst, it is not clar whthr. can b xpctd from.3. Firstly, our asymptotic is unconditional and scondly, on a mor tchnical lvl, th approximat functional uations ndd for.3 ar st up without Gauss sums [, sction 3] whil for., Gauss sums cannot b avoidd. Finally w rmark that th problm w ar considring is analytic in natur. For comparision, w mntion that Chinta [3] has shown that for a fixd lliptic curv E ovr Q and a larg prim, th cntral valu L/, E χ is non-zro for all but O 7/8+ɛ Dirichlt charactrs χ modulo. Givn this powrful rsult, th simultanous non-vanishing problms discussd abov ar almost trivial whn posd for wight two holomorphic Hck-cusp forms. For Hck-aass forms howvr, th algbraic mthods usd in [3] ar not applicabl... Acknowldgmnts. W ar gratful to Shng-Chi Liu for providing us with his prprint []. W thank th Dpartmnt of athmatics, Indian Institut of scinc, Bangalor and Txas A& Univrsity at Qatar, whr this work was don, for thir hospitalitis. Th first author acknowldgs th financial support by UGC-DST India during this work.. Prliminaris W giv th proof of Thorm. for vn Hck-aass forms, th dtails for odd forms bing ntirly similar. Thus throughout, will dnot a prim and f an vn Hck-aass cusp form for th full modular group with Laplacian ignvalu + T f, whr T f is ral. Lt λ f n dnot th ignvalu of th n-th Hck oprator corrsponding to f. For mor dtails on aass forms, w rfr th radr to [7]. At th tim this papr was submittd for publication, [, Thorm.] was conditional on th Ramanujan-Ptrsson conjctur, but now it is unconditional.

3 SIULTANEOUS NONVANISHING OF L-FUNCTIONS 3 For an vn Dirichlt charactr χ modulo, w dfin th L-functions. Ls, χ = χnn s, Ls, f χ = λ f nχnn s n= for Rs >, with analytic continuation to ntir functions having th functional uations. whr.3 Λs, χ := π Λs, χ = τχ Λ s, χ, s/ s Ls, χ, Λs, f χ := π n= Λs, f χ = τχ Λ s, f χ, s s + itf s itf Ls, f χ. W will us th convntion that ɛ dnots an arbitrarily small positiv constant, but not ncssarily th sam constant from on occurrnc to th nxt. All implid constants may dpnd implicitly on ɛ and f... Orthogonality of charactrs. W will nd th following basic idntity. This and mor on Dirichlt charactrs can b found in []. Lmma.. For prim and nm, =, w hav. χnχm = χ mod if n m mod othrwis... Approximat functional uations. W will nd th following xprssions for th cntral valus of Ls, χ and Ls, f χ. Ths can b drivd in a standard way from [8, Thorm 5.3] and th functional uations of ths L-functions. Lmma.. For χ an vn primitiv charactr of modulus, w hav L, χ = L, χ = χm m V + τχ χm m.5 V, m m whr for x, c > 0,.6 V x = πi c πx s s+ W hav that th l-th drivativ V l x l,c min{, x C l } for any C > 0, and V x for x < ɛ. Thus th sums in.5 ar ssntially supportd on m < /+ɛ. Lmma.3. For χ an vn primitiv charactr of modulus, w hav L, f χ = λ f nχn n V + τχ.7 n whr for x, c > 0,.8 V x = πx s πi c s++itf +itf ds s λ f nχn n V n s+ itf itf W hav that th l-th drivativ V l x l,c min{, x C l } for any C > 0, and V x for x < ɛ. Thus th sums in.7 ar ssntially supportd on n < +ɛ. ds s.,

4 SOUYA DAS AND RIZWANUR KHAN.3. Sums of Fourir cofficints. Th Ramanujan conjctur for th Fourir cofficints of f is tru on avrag, by Rankin-Slbrg thory. Namly, w hav.9 Individually, w hav Kim and Sarnak s [0] bound / λ f n x / λ f n x. n<x n<x.0 λ f n n θ+ɛ, whr θ = 7/6. W will also ncountr sums of Fourir cofficints twistd by additiv charactrs. In this contxt, lt us rcall th Voronoi summation formula, Lmma.. [5, Thorm.] Lt ψ b a fixd smooth function with compact support on th positiv rals. Lt d, d Z with, d = and dd mod. Thn nd n λ f n ψ = λ f n nd nn N n Ψ + + λ f n nd nn. n Ψ, whr for σ >,. ψs is th llin transform of ψx and +s+itf.3 πg ± s = Ψ ± x = π x s G ± s πi ψ sds, σ s+itf +s itf s itf ± +s+itf + s+itf + +s itf + s itf + Th following rsult says that Fourir cofficints ar orthogonal to additiv charactrs on avrag. Lmma.5. [7, Thorm 8.] For any ral numbr α, w hav that. λ f nαn N /+ɛ. n N Th implid constant dos not dpnd on α.. 3. Proof of Thorm. Using th approximat functional uations and by picking out vn charactrs using th factor χ + which uals whn χ = and 0 othrwis, w hav that th lft hand sid of. uals 3. χ mod which w writ as 3. χ + λ f nχn n V + ik τχ n χ mod χ + S + S S 3 + S. λ f nχn n V n χm m V + τχ χm m V. m m ultiplying out th summand abov lads to svral cross trms, which w now analyz on by on.

5 SIULTANEOUS NONVANISHING OF L-FUNCTIONS Cross trms with no Gauss sum. In this sction w considr 3.3 χ mod Lmma χ + S S 3 = χ mod n, χ mod n, λ f nχnχm m n V V nm + χ mod n, λ f nχ nχm m n V V 3/+θ+ɛ. nm Proof. By Lmma. and.0, it suffics to bound by 3/+θ+ɛ th sum λ f n 3.5 +θ+ɛ nm n< +ɛ,m< /+ɛ n+m 0 mod n< +ɛ,m< /+ɛ n+m 0 mod λ f nχ nχm m n V V. nm nm. Writing n = k m, w s from th rangs of n and m that w must hav k < ɛ. Thus 3.5 is boundd by 3.6 7/6+ɛ 3/+θ+ɛ. m m< /+ɛ Th nxt rsult givs th main trm of Thorm.. Lmma χ mod n, λ f nχnχm m n V V = nm L, f + O3/+θ+ɛ Proof. By Lmma. and.9, th lft hand sid of 3.7 uals λ f n m n 3.8 V V nm n, n m mod nm,= + O 3/+ɛ. By a similar argumnt as that usd to prov Lmma 3., w s that up to an rror of O 3/+θ+ɛ, th main trm in 3.8 consists of thos trms with n = m: λ f n n n n V V = λ f n n n 3.9 n V V + O 00. n,= Th uality abov holds bcaus V n is vry small unlss n < /+ɛ, in which cas n, = is automatic. Using th dfinitions of V and V, w gt λ f n n n 3.0 n V V = λ f n s+ s++it πi π s/ s +s+s n f s+ it f ds ds. s s +itf itf Th n-sum insid th intgral uals L + s + s, f. Shifting th lins of intgration to Rs = Rs = / + ɛ, w pick up th rsidu L, f at s = s = 0. Th intgral on th nw lins can b boundd in a standard way by /+ɛ. This complts th proof.

6 6 SOUYA DAS AND RIZWANUR KHAN 3.. Cross trms with on Gauss sum. In this sction w considr 3. Lmma χ mod χ mod χ mod χ + S S χ + τχ χ + and χ mod χ + S S. λ f nχn n V τχ n λ f nχn n V τχ χm m V 3/+ɛ. n m χm m V m 3/+ɛ, Proof. Th proofs of 3. and 3.3 ar similar, so w show only th formr. For vn primitiv charactrs w hav that τχτχ =. Thus to prov 3. w nd to bound by 3/+ɛ th sum λ f n m n 3. V / V τχχ±nχm. nm n, nm,= χ mod Writing τχ = a mod χaa/, th innrmost sum of 3. uals τχχ±nχm = a 3.5 χ±nχma. χ mod Now using Lmma., w hav that 3. uals m 3.6 V / m 3.7 / m,= m,= m V m a mod n,= n,= λ f n n χ mod ±nm n V λ f n n V n a mod a. Th innrmost a-sum of 3.7 is a Ramanujan sum which uals, so that 3.7 is trivially boundd by /+ɛ. As for 3.6, rmoving th condition n, = and using.0, w hav that it uals ±nm n 3.8 V / + O /+θ+ɛ. m,= m λ f n V m n Now by Lmma.5 and partial summation, w gt that th n-sum in 3.8 is boundd by ɛ, th m-sum by /. Hnc 3.8 is boundd by 3/+ɛ Cross trms with two Gauss sums. In this sction w considr 3.9 Lmma χ mod χ + τχ χ mod χ + S S 3. λ f nχn n V χm m V 7/8+θ+ɛ nm m Proof. By taking a smooth partition of unity, w may considr th n and m sums in dyadic intrvals. Thus it suffics to bound by 7/8+θ+ɛ th sum τχ m m n n 3. λ N / f nχ±nmv W V W, N χ mod n,

7 SIULTANEOUS NONVANISHING OF L-FUNCTIONS 7 for any fixd smooth functions W i x supportd on x [, ], N < +ɛ and < /+ɛ. Writing τχ = a mod χaa/, w hav that 3. uals 3. N / n, nm,= m m n n λ f nv W V W N a,b mod a + b χ mod χ±nmχab. Th innrmost sum can b valuatd using Lmma.. Thus 3. uals N / N / n, nm,= n, nm,= m m n n λ f nv W V W N a mod m m n n λ f nv W V W N a,b mod a ± nma a + b Th innrmost a, b-sum of 3. is a product of two Ramanujan sums and uals. Thus 3. is boundd absolutly by /+ɛ. As for 3.3, w procd according to th sizs of N and. Cas I: N < /, < /. W not that th innrmost a-sum of 3.3 is a Kloostrman sum, which is lss than / by Wil s bound. Using this and.9, w gt that 3.3 is boundd by 3.5 ɛ N / 7/8+ɛ. Cas II: /. W first not that th m-sum in 3.3 uals 3.6 m,= ±nma m m V W = ±nma m m V W + O 00, bcaus V m is vry small unlss m < /+ɛ, in which cas m, = is automatic. By Poisson summation, w hav that 3.7 m m V W ±nma = ±nba b = = b ±nba m= b mod m= b + m b + m V W b±na m b + x b + x V W mxdx y V W y my dy. Rpatd intgration by parts shows that th intgral abov is boundd by +ɛ / m B for any B 0. Thus w may rstrict th m-sum in th last lin to m < +ɛ /, up to an rror of 00 say. Th b-sum

8 8 SOUYA DAS AND RIZWANUR KHAN uals if ±na m mod, and 0 othrwis. Thus 3.3 is boundd by / ±nm n n 3.8 N / λ f n V W N 0< m < +ɛ / m,= n,= = / ±nm 3.9 N / λ f n 0< m < +ɛ / m,= + O 00 n n V W N + O/+θ+ɛ. Abov, 3.9 was obtaind from 3.8 by bounding absolutly th contributing of th intgrs n divisibl by, of which thr ar at most ɛ. By Lmma.5 and partial summation, w find that th n-sum in 3.9 is boundd by N /+ɛ and so 3.9 is boundd by ɛ / 7/8+ɛ. Cas III: N /, < /. By rmoving th condition n, =, w not that 3.3 uals m m a n n ±nma 3.3 V N / W λ f nv W N a mod m,= Applying Voronoi summation to th innrmost n-sum, w gt that 3.3 uals m m a 3.3 V N / W λ f n nma n a mod m,= plus a similar sum involving Ψ +, whr Ψ ± x = 3.33 π x s G ± s πi 0 0 V tn/w tt s dt ds. + O /+θ+ɛ. nn Ψ Th a-sum in 3.3 uals if n m mod and othrwis, so that 3.3 uals m m λ f n nn 3.3 V N / W n Ψ 3.35 N / m,= m,= V m W m n m mod W now xplain how to truncat th n-sum. W first not that 3.36 G ± s 0 λ f n n Ψ V tn/w tt s dt + s Rs+ nn + O /+θ+ɛ. ɛ + s B, + O /+θ+ɛ by Stirling s approximation for th gamma function and by intgrating by parts svral tims th t-intgral, for any B 0, whr th implid constant dpnds on Rs, B and of cours f. Using this stimat, by shifting th lin of intgration in 3.33 right to Rs = C, w hav that Ψ ± x ɛ x C for any C > 0. Thus w may rstrict 3.3 and 3.35 to n < +ɛ /N, up to an rror of O 00 say. Also, w may shift th lin of intgration in 3.33 lft to Rs = + ɛ to gt that that Ψ ± x ɛ x.

9 SIULTANEOUS NONVANISHING OF L-FUNCTIONS Now rstricting to n < +ɛ /N and bounding absolutly w find that that 3.3 is boundd by +ɛ N / m< n< +ɛ /N n m mod λ f n nn n +θ+ɛ / 7/8+θ+ɛ. N / Th sam bound holds for Sinc th sum involving Ψ + can b tratd in xactly th sam way, this complts th proof. Rfrncs. V. Blomr, É Fouvry, E. Kowalski, P. ichl, and D. ilićvić, On momnts of twistd L-functions, prprint.. Valntin Blomr and Djordj ilićvić, Th scond momnt of twistd modular L-functions, prprint. 3. Gautam Chinta, Analytic ranks of lliptic curvs ovr cyclotomic filds, J. Rin Angw. ath. 5 00, 3.. Harold Davnport, ultiplicativ numbr thory, third d., Graduat Txts in athmatics, vol. 7, Springr-Vrlag, Nw York, 000, Rvisd and with a prfac by Hugh L. ontgomry. 5. Danil Godbr, Additiv twists of Fourir cofficints of modular forms, J. Numbr Thory 33 03, no., Jff Hoffstin and in L, Scond momnts and simultanous non-vanishing of GL automorphic L-sris, prprint. 7. Hnryk Iwanic, Spctral mthods of automorphic forms, scond d., Graduat Studis in athmatics, vol. 53, Amrican athmatical Socity, Providnc, RI; Rvista atmática Ibroamricana, adrid, Hnryk Iwanic and Emmanul Kowalski, Analytic numbr thory, Amrican athmatical Socity Collouium Publications, vol. 53, Amrican athmatical Socity, Providnc, RI, Hnryk Iwanic and Ptr Sarnak, Th non-vanishing of cntral valus of automorphic L-functions and Landau-Sigl zros, Isral J. ath , no. part A, Hnry H. Kim, Functoriality for th xtrior suar of GL and th symmtric fourth of GL, J. Amr. ath. Soc , no., 39 83, With appndix by Dinakar Ramakrishnan and appndix by Kim and Ptr Sarnak.. Shng-Chi Liu, Simultanous nonvanishing of automorphic L-functions, prprint.. P. ichl and J. Vandrkam, Simultanous nonvanishing of twists of automorphic L-functions, Compositio ath. 3 00, no., Dpartmnt of athmatics, Indian Institut of Scinc, Bangalor, India. addrss: somu@math.iisc.rnt.in Scinc Program, Txas A& Univrsity at Qatar, Doha, Qatar addrss: rizwanur.khan@atar.tamu.du

Hardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.

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