Lower Bounds on Odd Order Character Sums

Transcription

1 Iteratioal Mathematics Research Notices Advace Access published November 28, 20 L. Goldmakher ad Y. Lamzouri 20) Lower Bouds o Odd Order Character Sums, Iteratioal Mathematics Research Notices, rr29, 8 pages. doi:0.093/imr/rr29 Lower Bouds o Odd Order Character Sums Leo Goldmakher ad Youess Lamzouri 2 Departmet of Mathematics, Uiversity of Toroto, 40 St. George Street, Toroto, ON, Caada M5S 2E4 ad 2 Departmet of Mathematics, Uiversity of Illiois at Urbaa-Champaig, 409 W. Gree Street, Urbaa, IL 680, USA Correspodece to be set to: lamzouri@math.uiuc.edu A classical result of Paley shows that there are ifiitely may quadratic characters χmod q) whose character sums get as large as q log log q; this implies that a coditioal upper boud of Motgomery ad Vaugha caot be improved. I this paper, we derive aalogous lower bouds o character sums for characters of odd order, which are best possible i view of the correspodig coditioal upper bouds recetly obtaied by the first author. Itroductio I this paper, we obtai lower bouds o the quatity Mχ) := t χ) t Dowloaded from at Biology Library o May, 202 for Dirichlet characters χmod q) of odd order. The study of character sums ad associated quatities such as Mχ) has bee a cetral topic of aalytic umber theory for This work begu while both authors were participatig i the AMS Mathematics Research Commuity The Pretetious View of Aalytic Number Theory held o Jue 26 July 2, 20 at the Sowbird Resort, Utah. Received September 6, 20; Accepted October 24, 20 c The Authors) 20. Published by Oxford Uiversity Press. All rights reserved. For permissios, please jourals.permissios@oup.com.

2 2 L. Goldmakher ad Y. Lamzouri a log time. The first result i this area, discovered idepedetly by Pólya ad Viogradov i 98, asserts that Mχ) q log q holds uiformly over all characters χmod q). This boud has resisted ay improvemet outside special cases. However, coditioally o the Geeralized Riema Hypothesis, Motgomery ad Vaugha [5] were able to improve this to Mχ) q log log q. For quadratic characters, this is kow to be optimal, owig to a ucoditioal lower boud due to Paley [7]. Furthermore, assumig the GRH, Graville ad Soudararaja [3] have exteded Paley s lower boud to characters of all eve orders. The story took a uexpected tur whe Graville ad Soudararaja discovered that both the Pólya Viogradov ad the Motgomery Vaugha bouds ca be improved for characters of odd order. I [3], they showed that for all characters χmod q) of odd order g 3, there exists δ g > 0 such that Mχ) g qlog q) δ g +o) ucoditioally ad Mχ) g qlog log q) δ g +o) coditioally o GRH. Here o) 0asq.) After developig their ideas further, Goldmakher [2] showed that these bouds hold with δ g = g π si π g. Dowloaded from at Biology Library o May, 202 Moreover, coditioally o GRH, he proved that this value is optimal. To be precise, o GRH he showed that for ay ɛ>0 ad ay fixed odd iteger g 3, there exist arbitrarily large q ad primitive characters χmod q) of order g satisfyig Mχ) g,ɛ qlog log q) δ g ɛ..) The goal of the preset article is to establish the same result ucoditioally.

3 Lower Bouds o Odd Order Character Sums 3 Recet progress o character sums was made possible by Graville ad Soudararaja s discovery that Mχ) depeds o the extet to which χ mimics the behavior of other characters. To measure this mimicry, they itroduced the symbol Dχ, ψ; y) := p y Re χp)ψp) p /2. It turs out that this defies a pseudometric o the space of characters, ad has a umber of iterestig properties; see [4] for a i-depth discussio. I [3], Graville ad Soudararaja derive a umber of upper ad lower bouds o Mχ) i terms of Dχ, ψ; y), where ψ is a character of small coductor ad opposite parity to χ i.e., χ )ψ ) = ). For example, they prove the followig theorem. Theorem A [3, Theorem 2.5]). Assume GRH. Let χmod q) ad ψ mod m be primitive characters with ψ ) = χ ). The Mχ) + qm qm log log log q φm) φm) log log q exp Dχ, ψ; log q)2 ). I this way, the problem of boudig Mχ) is traslated ito that of boudig Dχ, ψ; log q). A cosequece of [3, Lemma 3.2] is that wheever χmod q) is a primitive character of odd order g ad ξ mod m is a primitive character of opposite parity ad small coductor m log log q) A ), Dχ, ξ; log q) 2 δ g + o)) log log log q. Goldmakher, usig a reciprocity law for the gth-order residue symbol ad the Chiese Remaider Theorem, proved a complemetary result: Dowloaded from at Biology Library o May, 202 Theorem B [2, Sectio 9]). Let g 3 be a fixed odd iteger. For ay ɛ>0, there exists a odd character ξ mod m with m ɛ ad a ifiite family of primitive characters χmod q) of order g such that Dχ, ξ; log q) 2 δ g + ɛ)log log log q. Combiig Theorems A ad B produces the lower boud.) o coditioally, sice Theorem A is depedet o the GRH. Mχ);

4 4 L. Goldmakher ad Y. Lamzouri To prove.) ucoditioally, we derive a ucoditioal versio of Theorem A. Although we caot prove a totally aalogous theorem, i the case where χ is eve we are able to remove the assumptio of GRH at a cost of a extra log log log q) factor: Theorem. Let χmod q) be a primitive eve character ad ψ mod m be a primitive odd character. The Mχ) + q qm log log q φm) log log log q exp Dχ, ψ; log q)2 ). Theorem 2. Combiig Theorem with Theorem B, we deduce the desired lower boud. Let g 3 be a fixed odd iteger. There exist arbitrarily large q ad primitive characters χmod q) of order g such that where δ g = g π si π g. 2 The Key Lemmas Mχ) g,ɛ qlog log q) δ g ɛ, I this sectio, we prove two geeral results that will be the mai igrediets i the proof of Theorem, ad might also be of idepedet iterest. Lemma 2.. Let f be a multiplicative fuctio such that f) for all, ad let y be a large real umber. The N y N f) + log log y log y log log y exp D f, ; y)2 ). Dowloaded from at Biology Library o May, 202 Remark. This lemma is a geeralizatio of [3, Lemma 6.3], where y is a positive iteger ad f is a primitive character modulo y. Note that i this special case, the factor log log y) ca be removed from the RHS of the earlier-metioed iequality. Proof. Let δ>0 be a real umber to be chose later. The y δ t +δ t f) dt = y f) y δ f) t dt = +δ +δ y δ y y f).

5 Hece, usig that y δ dt = y δ, we derive t +δ Lower Bouds o Odd Order Character Sums 5 f) +δ y N y N f). 2.) For Res)>, let F s) = = f)/s be the Dirichlet series of f. The F + δ) = y f) + O +δ >y ) = +δ y ) f) + O. +δ δy δ We choose δ = log log y/ log y, which yields F + δ) = y f) + O +δ log log y O the other had, we obtai from the Euler product of F s) log F + δ) = p Therefore, we get which implies log F + δ) p y fp) + O) = p+δ p e /δ Re fp) p e /δ <p y fp) + O) = p+δ p e /δ ). 2.2) fp) p + O). + O) = log log log y + O), p F + δ) log y log log y exp D f, ; y)2 ). Dowloaded from at Biology Library o May, 202 The lemma follows upo combiig this last estimate with 2.) ad2.2). Our secod lemma is ispired by Paley s approach i [7]. Recall that the Fejér kerel, defied by F N θ) := N ) eθ), N

6 6 L. Goldmakher ad Y. Lamzouri where et) = e 2πit,satisfies F N θ) = N ) siπ Nθ) 2 0 ad siπθ) 0 F N θ) dθ =. Usig these properties of the Fejér kerel, we establish the followig lemma: Lemma 2.2. Let {a)} Z be a sequece of complex umbers with a) for all, ad let x 2 be a real umber. The Proof. θ [0,] N x N a) eθ) = θ [0,] x a) eθ) + O). To establish the result, we eed to prove oly the implicit upper boud, sice the implicit lower boud holds trivially. Let α [0, ], ad N x. First, ote that Moreover, we have N N a) eα) = N a) eα) ) + O). N a) eα) ) = a) N eα) m ) emθ)e θ)dθ N x m N 0 = a) eα θ)) F N θ) dθ. 0 x Dowloaded from at Biology Library o May, 202 Thus, we obtai N a) eα) θ [0,] x a) eθ) + O). Sice α ad N were arbitrary the result follows.

7 Lower Bouds o Odd Order Character Sums 7 3 Proof of Theorem Give a primitive character χmod q), recall Pólya s fourier expasio [6]: t χ) = τχ) 2πi q χ) e t )) + Olog q), q where τχ):= χb)eb/q) b mod q) is the Gauss sum. It follows that if χ is eve, t χ) = τχ) 2πi ad therefore we get i this case Proof of Theorem. Now, usig that θ [0,] q q Mχ) + log q q θ [0,] χ) e t ) + Olog q), q q First, we ifer from Lemma 2.2 that χ) eθ) = b mod m) θ [0,] N q N ψb)eb/m) = ψ)τψ), χ) eθ). 3.) χ) eθ) + O). 3.2) Dowloaded from at Biology Library o May, 202 for all itegers sice ψ is primitive, see e.g., []) we get b mod m) ψb) N ) χ) b e = ψb) m b mod m) N = 2τψ) N χ) ψ), χ) ) b b e e m m ))

8 8 L. Goldmakher ad Y. Lamzouri sice ψ ) =. We therefore obtai θ [0,] N χ) eθ) m φm) Combiig this estimate with 3.) ad3.2) we deduce Mχ) + qm q φm) N q N N χ) ψ) χ) ψ).. Fially, appealig to Lemma 2., we fid N q N χ) ψ) which completes the proof. Fudig + N log q N χ) ψ) + log log q log log log q exp Dχ, ψ; log q)2 ), The secod author is supported by a postdoctoral fellowship from the Natural Scieces ad Egieerig Research Coucil of Caada. Refereces [] Daveport, H. Multiplicative Number Theory. Graduate Texts i Mathematics 74. New York: Spriger, [2] Goldmakher, L. Multiplicative mimicry ad improvemets of the Polya Viogradov iequality. Algebra ad Number Theory, preprit arxiv: [3] Graville, A. ad K. Soudararaja. Large character sums: pretetious characters ad the Pólya Viogradov theorem. Joural of the America Mathematical Society 20, o ): [4] Graville, A. ad K. Soudararaja. Pretetious Multiplicative Fuctios ad a Iequality for the Zeta-Fuctio. Aatomy of Itegers, CRM Proceedigs & Lecture Notes 46. Providece, RI: America Mathematical Society, [5] Motgomery, H. L. ad R. C. Vaugha. Expoetial sums with multiplicative coefficiets. Ivetioes Mathematicae 43, o. 977): [6] Motgomery, H. L. ad R. C. Vaugha. Multiplicative Number Theory I. Classical Theory. Cambridge Studies i Advaced Mathematics 97. Cambridge: Cambridge Uiversity Press, [7] Paley, R. E. A. C. A theorem o characters. Joural of the Lodo Mathematical Society 7, o. 932): Dowloaded from at Biology Library o May, 202