LARGE ODD ORDER CHARACTER SUMS AND IMPROVEMENTS OF THE PÓLYA-VINOGRADOV INEQUALITY

Transcription

1 LARGE ODD ORDER CHARACTER SUMS AND IMPROVEMENTS OF THE PÓLYA-VINOGRADOV INEQUALITY YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Abstract For a rimitive Dirichlet character χ modulo q, we defie Mχ = max t t χ I this aer, we study this quatity for characters of a fixed odd order g 3 Our mai result rovides a further imrovemet of the classical Pólya-Viogradov iequality i this case More secifically, we show that for ay such character χ we have Mχ ε qlog q δ g log log q /4+ε, where δ g := g π siπ/g This imroves uo the works of Graville ad Soudararaja ad of Goldmakher Furthermore, assumig the Geeralized Riema hyothesis GRH we rove that Mχ q log 2 q δg log 3 q 4 log 4 q O, where log j is the j-th iterated logarithm We also show ucoditioally that this boud is best ossible u to a ower of log 4 q Oe of the key igrediets i the roof of the uer bouds is a ew Halász-tye iequality for logarithmic mea values of comletely multilicative fuctios, which might be of ideedet iterest Itroductio The study of Dirichlet characters ad their sums has bee a cetral toic i aalytic umber theory for a log time Let q 2 ad χ be a o-ricial Dirichlet character modulo q A imortat quatity associated to χ is Mχ := max t q χ The best-kow uer boud for Mχ, obtaied ideedetly by Pólya ad Viogradov i 98, reads t Mχ q log q Though oe ca establish this iequality usig oly basic Fourier aalysis, imrovig o it has roved to be a difficult roblem, ad resisted substatial rogress for several decades Coditioally o the Geeralized Riema Hyothesis GRH, Motgomery ad Vaugha [5] showed i 977 that 2 Mχ q log log q 200 Mathematics Subject Classificatio Primary L40 The first author is artially suorted by a Discovery Grat from the Natural Scieces ad Egieerig Research Coucil of Caada

2 2 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL This boud is best ossible i view of a old result of Paley [6] that there exists a ifiite family of rimitive quadratic characters χ mod q such that 3 Mχ q log log q Assumig GRH, Graville ad Soudararaja [9] exteded Paley s result to characters of a fixed eve order 2k 4 The assumtio of GRH was later removed by Goldmakher ad Lamzouri [6], who obtaied this result ucoditioally, ad subsequetly Lamzouri [0] obtaied the otimal imlicit costat i 3 for eve order characters The situatio is quite differet for odd order characters I this case, Graville ad Soudararaja [9] roved the remarkable result that both the Pólya-Viogradov ad the Motgomery-Vaugha bouds ca be imroved More secifically, if g 3 is a odd iteger, ad χ is a rimitive character of order g ad coductor q the they showed that 4 Mχ qlog Q δg 2 +o, where δ g := g siπ/g ad π { q 5 Q := log q ucoditioally, o GRH By refiig their method, Goldmakher [4] was able to obtai the imroved boud 6 Mχ qlog Q δg+o Our first result gives a further imrovemet of the Pólya-Viogradov iequality for Mχ whe χ has odd order g 3 Here ad throughout, we write log k x = loglog k x to deote the kth iterated logarithm, where log x = log x Theorem Let g 3 be a fixed odd iteger, ad let ε > 0 be small The, for ay rimitive Dirichlet character χ of order g ad coductor q we have Mχ ε q log q δ g log log q 4 +ε The occurrece of ε i the exoet of log log q i the uer boud is a cosequece of the ossible existece of Siegel zeros I articular, if Siegel zeros do ot exist the the log log q ε term ca be relaced by log 3 q O Assumig GRH, ad usig results of Graville ad Soudararaja see Theorem 24 below, Goldmakher [4] also showed that the coditioal boud i 6 is best ossible More recisely, for every ε > 0 ad odd iteger g 3, he roved the existece of a ifiite family of rimitive characters χ mod q of order g such that 7 Mχ ε qlog log q δ g ε, coditioally o the GRH By modifyig the argumet of Graville ad Soudararaja ad usig ideas of Paley [6], Goldmakher ad Lamzouri [5] roved this result ucoditioally

3 LARGE ODD ORDER CHARACTER SUMS 3 It is atural to ask to what degree of recisio we ca determie the exact order of magitude of the maximal values of Mχ whe χ has odd order g 3; i articular, ca we determie the otimal log log q o cotributios i the coditioal art of 6, ad i 7 We make rogress i this directio by showig that this term ca be relaced by log 3 q 4 log 4 q O i both 6 ad 7 This allows us to coditioally determie the maximal values of Mχ, u to a ower of log 4 q Theorem 2 Assume GRH Let g 3 be a fixed odd iteger The for ay rimitive Dirichlet character χ of order g ad coductor q we have 8 Mχ q log 2 q δg log 3 q 4 log4 q O Theorem 3 Let g 3 be a fixed odd iteger There are arbitrarily large q ad rimitive Dirichlet characters χ modulo q of order g such that 9 Mχ q log 2 q δg log 3 q 4 log4 q O To obtai Theorem 3, our argumet relates Mχ to the values of certai associated Dirichlet L-fuctios at, ad uses zero-desity results ad ideas from [0] to costruct characters χ for which these values are large We shall discuss the differet igrediets i the roofs of Theorems, 2 ad 3 i detail i the ext sectio Recet rogress o character sums was made ossible by Graville ad Soudararaja s discovery of a hidde structure amog the characters χ havig large Mχ I articular, they show that Mχ is large oly whe χ reteds to be a character of small coductor ad oosite arity To defie this otio of retetiousess, we eed some otatio Here ad throughout we deote by F the class of comletely multilicative fuctios f such that f for all For f, g F we defie 2 Refg Df, g; y :=, y which turs out to be a seudo-metric o F see [9] We say that f reteds to be g u to y if there is a costat 0 δ < such that Df, g; y 2 δ log log y Oe of the key igrediets i the roof of 4 is the followig boud for logarithmic mea values of fuctios f F i terms of Df, ; x see Lemma 43 of [9] f 0 log x ex 2 Df, ; x2 x Note that the factor /2 iside the exoetial o the right had side of 0 is resosible for the weaker exoet δ g /2 i 4 Goldmakher [4] realized that oe ca obtai the otimal exoet δ g i 6 by relacig 0 by a Halász-tye iequality for logarithmic mea values of multilicative fuctios due to Motgomery ad Vaugha [3] Combiig Theorem 2 of [3] with

4 4 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL refiemets of Teebaum see Chater III4 of [7] he deduced that see Theorem 24 i [4] f log x ex Mf; x, T +, T x for all f F ad T, where Mf; x, T := mi t T Df, it ; x 2 Motivated by our ivestigatio of character sums, we are iterested i characterizig the fuctios f F that have a large logarithmic mea, i the sese that f 2 log xα, x x for some 0 < α Takig T = i shows that this haes oly whe f reteds to be it for some t However, observe that it = xit 2 it + O mi it t, log x, ad hece f = it satisfies 2 oly whe t log x α By refiig the ideas of Motgomery ad Vaugha [3] ad Teebaum [7], we rove the followig result, which shows that this is essetially the oly case Theorem 4 Let f F ad x 2 The, for ay real umber 0 < T we have f log x ex Mf; x, T + T, x where the imlicit costat is absolute Takig T = clog x α i this result where c > 0 is a suitably small costat, we deduce that if f F satisfies 2, the f reteds to be it for some t log x α Theorem 4 will be oe of the key igrediets i obtaiig our suerior bouds for Mχ i Theorems ad 2 2 Detailed statemet of results To exlai the key ideas i the roofs of Theorems, 2 ad 3, we shall first sketch the argumet of Graville ad Soudararaja [9] Their startig oit is Pólya s Fourier exasio see sectio 94 of [4] for the character sum t χ, which reads 2 χ = τχ χ e t + O + q log q, 2πi q N t N where χ is a rimitive character modulo q, ex := e 2πix ad τχ is the Gauss sum q τχ := χe q =

5 LARGE ODD ORDER CHARACTER SUMS 5 Note that τχ = q wheever χ is rimitive Thus, i order to estimate Mχ, oe eeds to uderstad the size of the exoetial sum 22 q χ eθ, for θ [0, ] Motgomery ad Vaugha [5] showed that this sum is small if θ belogs to a mior arc, ie, θ ca oly be well-aroximated by ratioals with large deomiators comared to q This leaves the more difficult case of θ lyig i a major arc I this case, θ ca be well-aroximated by some ratioal b/r with suitably small r comared to q Graville ad Soudararaja showed that i this case there is some large N deedig o θ, b, r ad q such that we ca aroximate the sum 22 by N = φr χ eb/r = ψ mod r a mod r a mod r eab/r ψaeab/r N a mod r N χ χψ The bracketed term, a Gauss sum, is well uderstood; i articular it has orm r, where r is the coductor of ψ see eg, Theorem 97 of [4] Thus, what remais to be determied i order to boud Mχ, is a uer boud for the sum 23 N χψ for each character ψ modulo r Furthermore, observe that if χ ad ψ have the same arity the this sum is exactly 0; hece, we oly eed to cosider the case whe χ ad ψ have oosite arities Graville ad Soudararaja s breakthrough stems from their discovery of a reulsio heomeo betwee characters χ of odd order which are ecessarily of eve arity, ad characters ψ of odd arity ad small coductor A cosequece of this heomeo is that the sum 23 is small, allowig them to imrove the Pólya-Viogradov iequality i this case More secifically, they show that if χ is a rimitive character of odd order g 3 ad ψ is a odd rimitive character of coductor m log y A the 24 Dχ, ψ; y 2 δ g + o log log y see Lemma 32 of [9] Isertig this boud i 0 allows them to boud the sum 23, from which they deduce the ucoditioal case of 4 The roof of the coditioal art of 4 whe Q = log q roceeds alog the same lies, but uses a additioal igrediet, amely the followig aroximatio for the sum 22 see Proositio

6 6 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL 23 ad Lemma 52 of [9] coditioal o GRH: χ 25 eθ = χ eθ + O y /6 log q 2 q q Sy Here, Sy is the set of y-friable itegers also kow as y-smooth itegers, ie, the set of ositive itegers whose rime factors are all less tha or equal to y I [4], Goldmakher showed that the boud 24 is best ossible Furthermore, i order to obtai the exoet δ g i 6, he used the iequality to boud the sum 23 i terms of Mχψ; y, T However, to esure that this argumet works, oe eeds to show that the lower boud 24 still ersists if we twist χψ by Archimedea characters it for t T By a careful aalysis of Mχψ; y, T, Goldmakher see Theorem 20 of [4] roved that uder the same assumtios as Mχψ; y, log y 2 δ g + o log log y Thus, by combiig this boud with ad followig closely the argumet i [9], he was able to obtai 6 I order to imrove these results ad establish Theorems ad 2, the first ste is to obtai more recise estimates for the quatity Mχψ; y, T We discover that there is a substatial differece betwee the sizes of Mχψ; y, T ad Mχψ; y, T 2 if T is small ad T 2 is large a result that may be surrisig i view of 24 ad 26 I fact, we rove that there is a large secodary term of size log 2 y/k 2 where k is the order of ψ that aears i the estimate of Mχψ; y, T whe T log y c for some costat c > 0, but disaears whe T Proositio 2 Let g 3 be a fixed odd iteger, α 0,, ad ε > 0 be small Let χ be a rimitive character of order g ad coductor q Let ψ be a odd rimitive character modulo m, with m log y 4α/7 Put k := k/k, g The we have 27 Mχψ; y, log y α δ g + απ2 δ g log 4gk 2 2 y βε log m + O α log 2 m, where β = if m is a excetioal modulus ad β = 0 otherwise Proositio 22 Assume GRH Let g 3 be a fixed odd iteger Let N be large, ad y log N/0 Let ψ be a odd rimitive character of coductor m such that ex 2 log 3 y m ex log y The, there exist at least N rimitive characters χ of order g ad coductor q N, such that for all T we have Mχψ; y, T δ g log 2 y + O log 2 m The secodary term of size log 2 y/k 2 i the right had side of 27 is resosible for the additioal savig of log 2 Q /4 where Q is defied i 5 i Theorems ad 2; clearly, it does ot aear i Proositio 22, eve i the rage m log 2 y 2 ε

7 LARGE ODD ORDER CHARACTER SUMS 7 i which this secodary term is large Note that whe m is a excetioal modulus see the recise defiitio i 4 below, there is a additioal term that aears whe estimatig Mχψ; y, log y α that has size log L, χ m, where χ m is the excetioal character modulo m I this case, the extra term ε log m o the right had side of 27 is due to Siegel s boud L, χ m ε m ε To comlete the roofs of Theorems ad 2, we shall use our Theorem 4 to boud the sum 23, where we might choose T = log y α to take advatage of Proositio 2 Note that i view of Proositio 22, oe loses the additioal savig of log 2 Q /4 i Theorems ad 2 if oe simly uses with T = log y 2, as i [4] By usig Theorem 4 ad followig the ideas i [9], we rove the followig result, which is a refiemet of Theorem 29 i [4] This together with Proositio 2 imlies both Theorems ad 2 Theorem 23 Let χ be a rimitive character modulo q, ad let Q be as i 5 Of all rimitive characters with coductor below log Q 4/, let ξ modulo m be that character for which M χξ; Q, log Q 7/ is a miimum The we have qm Mχ χ ξ φm log Q ex M χξ; Q, log Q 7 + q log Q 9 +o Note that δ g is decreasig as a fuctio of g, so δ g δ > 9/ for all g 3 Therefore, whe χ is a rimitive character of odd order g 3 ad coductor q, we get the better boud Mχ q log Q 9 +o, uless ξ is odd ad M χξ; Q, log Q 7 is small We ext discuss the ideas that go ito the roof of Theorem 3 To obtai 7 uder GRH, Goldmakher [4] used the followig result from [9], which relates Mχ to the distace betwee χ ad ay rimitive character ψ with small coductor ad arity oosite to that of χ Theorem 24 Theorem 25 of [9] Assume GRH Let χ mod q ad ψ mod m be rimitive characters such that χ = ψ The we have qm qm Mχ + φm log 3 q φm log 2 q ex Dχ, ψ; log q 2 Thus, it oly remais to roduce characters χ ad ψ which satisfy the assumtios of Theorem 24, ad for which the lower boud 24 is attaied whe y = log q Usig the Eisestei recirocity law, Goldmakher see Proositio 93 of [4] roved that for ay ε > 0, there exists a odd rimitive character ψ modulo m ε, ad a ifiite family of rimitive characters χ mod q of order g such that 28 Dχ, ψ; log q 2 δ g + ε log 3 q To remove the assumtio of GRH, Goldmakher ad Lamzouri [5] see Theorem of [5] used ideas of Paley [6] to obtai a weaker versio of Theorem 24 ucoditioally

8 8 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Namely, they showed that if χ is odd ad ψ is eve the Mχ + qm log2 q q ex Dχ, ψ; log q 2 φm log 3 q Although this boud is eough to obtai 7 ucoditioally i view of 28, it is ot sufficiet to yield the recise estimate i Theorem 3, due to the loss of a factor of log 3 q over Theorem 24 Usig a comletely differet method, based o zero desity estimates for Dirichlet L-fuctios, we recover the origial boud of Graville ad Soudararaja ucoditioally for all characters χ modulo q with q N, excet for a small excetioal set of cardiality N ε Our argumet also gives a simle roof of Theorem 24, which exloits the atural roerties of the values of Dirichlet L-fuctios at, ad avoids the difficult study of exoetial sums with multilicative fuctios see Sectio 6 of [9] Note that the statemet of Theorem 24 trivially holds whe m > log q, sice Dχ, ψ; log q 2 log 3 q We thus oly eed to cosider the case m log q Theorem 25 Let ε > 0 ad let N be large Let m log N be a ositive iteger ad let ψ be a rimitive character modulo m The, for all but at most N ε rimitive characters χ modulo q with q N ad such that χ = ψ we have 29 Mχ + q ε qm φm log 2 q ex Dχ, ψ; log q 2 Moreover, if we assume GRH, the 29 is valid for all rimitive characters χ modulo q with q N, ad the imlicit costat i 29 is absolute To comlete the roof of Theorem 3, we thus eed to refie the estimate 28, ad this ca be achieved usig the same ideas as i the roof of Proositio 2 However, Goldmakher s roof of 28 oly roduces a ifiite sequece of rimitive characters χ, ad this is ot eough to use i Theorem 25, due to the ossible existece of a excetioal set of characters for which 29 does ot hold To overcome this difficulty, we use the results of [0] to rove the existece of may rimitive characters χ of order g ad coductor q N such that whe y log N, Dχ, ψ; y is maximal Proositio 26 Let g 3 be a fixed odd iteger Let N be large ad y log N/0 be a real umber Let m be a o-excetioal modulus such that m log y 4/7, ad let ψ be a odd rimitive character of coductor m Let k be the order of ψ ad ut k = k/g, k The, there exist at least N rimitive characters χ of order g ad coductor q N such that 20 Dχ, ψ; y 2 π/gk = δ g log taπ/gk 2 y + O log 2 m

9 LARGE ODD ORDER CHARACTER SUMS 9 3 A lower boud for Mχ: Proof of Theorem 25 The mai igrediet i the roof of Theorem 24 of [9] is the aroximatio 25, which is valid uder the assumtio of GRH To avoid this assumtio, we shall istead relate Mχ to the values of certai Dirichlet L-fuctios at s =, ad the use the classical zero-desity estimates for these L-fuctios Proositio 3 Let q be large ad m q/log q 2 Let χ mod q ad ψ mod m be rimitive characters such that ψ = χ The we have Mχ + qm q φm L, χψ We first eed the followig lemma Lemma 32 Let q be large ad m q/log q 2 Let χ be a character modulo q ad ψ be a character modulo m such that χψ is o-ricial The L, χψ = q χψ + O Proof Note that χψ is a o-ricial character of coductor at most qm q/ log q 2 Therefore, usig artial summatio ad the Pólya-Viogradov iequality we obtai χψ = χψk + O, + q< N ad the claim follows q< N q<k Proof of Proositio 3 Takig N = q i 2 gives Mχ + log q q max χ θ e θ Moreover, we observe that b mod m ψb q χ q b e = χ m q = τψ q b mod m ψbe χψ, b m which follows from the idetity b mod m ψbe b = ψτψ m

10 0 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Sice χ ad ψ are rimitive ad m q/log q 2 the χψ is o-ricial Therefore, by Lemma 32 together with the fact that χψ = we deduce that q χψ = 2 The result follows uo otig that ψb χ b mod m q q b e m χψ φm max θ = 2L, χψ + O q χ e θ, ad that τψ = m by the rimitivity of ψ I order to comlete the roof of Theorem 25, we eed to aroximate L, χψ by a short trucatio of its Euler roduct Usig zero desity estimates, we rove that this is ossible for almost all rimitive characters χ Proositio 33 Fix 0 < ε < ad let A = 00/ε Let N be large ad m log N The for all but at most N ε rimitive characters χ modulo q N we have 3 L, χψ = + O χψ log N log A N for all rimitive characters ψ modulo m Moreover, if we assume GRH, the 3 is valid with A = 0, for all rimitive characters χ modulo q N ad ψ modulo m I order to rove this roositio, we first eed some relimiary results Lemma 34 Let q be large ad χ be a o-ricial character modulo q Let 2 T q 2 ad X 2 Let σ 2 0 < ad suose that the rectagle {s : σ 0 < Res, Ims T + 3} does ot cotai ay zeros of Ls, χ The we have log L, χ = log χ log X + O + log q log q log T + T σ 0 T σ 0 2 Xσ 0 /2 X Proof Let α = / log X The it follows from Perro s formula that 32 2πi α+it α it log L + s, χ Xs s ds = Λ log χ + O X = = Λ log X log χ + O T X Λ +α log mi + X,, T log X/

11 LARGE ODD ORDER CHARACTER SUMS by a stadard estimatio of the error term Moreover, we observe that Λ log χ = log χ + O k k X X k=2 k >X = log χ + O X 2 X We ow move the cotour i 32 to the lie Res = σ, where σ = + σ 0 /2 We ecouter a simle ole at s = 0 that leaves a residue of log L, χ Furthermore, it follows from Lemma 8 of [7] that for σ σ ad t T we have Therefore, we deduce that where 2πi log Lσ + it, χ α+it α it σ it E = 2πi log q σ 0 T α it log q σ σ 0 log q σ 0 log L + s, χ Xs ds = log L, χ + E, s σ +it α+it + + log L + s, χ Xs σ it σ +it s ds + log q log T σ 0 2 Xσ 0 /2 Sice σ 0 /2, combiig the above estimates comletes the roof Lemma 35 Let ξ mod q ad ψ mod m be rimitive characters The, there is a uique rimitive character χ such that χψ is iduced by ξ if m q, ad o such character exists if m q Proof Suose that χψ is iduced by ξ, where χ is a rimitive character of coductor l The we must have q = [l, m], ad hece there is o such character χ if m q Now, suose that m q, ad let m = a a k k be its rime factorizatio We costruct χ i this case as follows Sice q = [l, m], the we have q = q 0 b b k k where q 0, m = ad b j a j for all j k, ad l = q 0 c c k k where c j = b j if b j > a j ad 0 c j a j if b j = a j Now, sice ξ is rimitive the ξ = ξ ξ ξ k where ξ is a rimitive character modulo q 0 ad ξ j is a rimitive character modulo b j j for j k Similarly, we have ψ = ψ ψ k ad χ = χ χ χ k where χ is a rimitive character modulo q 0 ad ψ j, χ j are rimitive characters modulo a j j ad c j j resectively Moreover, sice ξ iduces χψ the we must have χ = ξ, ad ξ j iduces χ j ψ j for all j k But this imlies that χ j = ξ j ψ j for all such that j, ad hece we deduce that there is oly oe choice for χ j sice it is rimitive Sice this holds for all j k, the character χ is uique

12 2 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Proof of Proositio 33 By Bombieri s classical zero-desity estimate see Theorem 20 of [2], we kow that there are at most N 6 σ log N B rimitive characters ξ with coductor q N log N ad such that Ls, ξ has a zero i the rectagle {s : σ Res, Ims N}, where B is a absolute costat Let ξ,, ξ L be these characters with σ = ε/20 The, it follows from the above argumet that L N ε/2 Recall that if ξ is a rimitive character that iduces ξ, the Ls, ξ ad Ls, ξ have the same zeros i the half-lae Res > 0 For a rimitive character ψ modulo m, let E ψ deote the set of rimitive characters χ modulo q with q N ad such that χψ is iduced by oe of the characters ξ j for j L Let E m be the uio over all rimitive characters ψ modulo m of the sets E ψ The, it follows from Lemma 35 that E m ψ mod m ψ rimitive E ψ Lφm N ε Let X = log N A where A = 00/ε If χ is a rimitive character with coductor q N ad such that χ / E m the it follows from Lemma 34 with T = X that for all rimitive characters ψ modulo m we have log L, χψ = X log χψ + O, log N which imlies 3 Fially, if we assume GRH, the this estimate is valid for all rimitive characters χ modulo q N ad ψ modulo m with X = log N 0 by Lemma 34 We ca ow rove Theorem 25 Proof of Theorem 25 Combiig Proositios 3 ad 33 we deduce that for all but at most N ε rimitive characters χ modulo q with N ε/3 q N we have 33 Mχ + q qm φm log A N χψ with A = 00/ε The first art of the theorem follows, uo otig that log A N χψ ε log 2 q ex Dχ, ψ; log q 2 The secod art follows alog the same lies, sice if we assume GRH the 33 holds with A = 0 for all rimitive characters χ with coductor q N

13 LARGE ODD ORDER CHARACTER SUMS 3 4 Estimates for the distace Dχ, ψ; y: Proofs of Proositio 26 ad Theorem 3 We shall first rove a lower boud for Dχ, ψ; y 2, which is a refied versio of 24, that shows that Proositio 26 is best ossible This will also be the mai igrediet i the roof of Proositio 2 Proositio 4 Let g 3 be a fixed odd iteger, ad ε > 0 be small Let ψ be a odd rimitive character of coductor m ad order k, ad y be such that m log y 4/7 Put k = k/g, k The, for ay rimitive character χ mod q of order g we have Dχ, ψ; y 2 π/gk δ g log taπ/gk 2 y βε log m + O log 2 m, where β = 0 if m is a o-excetioal modulus, ad β = if m is excetioal We say that m is a excetioal modulus if there exists a Dirichlet character χ m ad a comlex umber s such that Ls, χ m = 0 ad 4 Res c logmims + 2 for some sufficietly small costat c > 0 Oe exects that there are o such moduli, but what is kow ucoditioally is that if m is excetioal, the there is oly oe excetioal character χ m modulo m, which is quadratic, ad for which Ls, χ m has a uique zero i the regio 4 which is real ad simle this zero is called a Siegel zero For g 3, we let µ g deote the set of g-th roots of uity The, we observe that Dχ, ψ; y 2 = log log y Reχψ + O y log log y max Re z e l 42 z µ g {0} k + O l mod k y ψ=e k l Proositio 4 follows from this iequality together with Proositio 42 below, which rovides a asymtotic formula for the sum o the right had side of 42 To establish Proositio 26, we eed a additioal igrediet, amely that there exist may rimitive characters χ whose values we ca cotrol at the small rimes c log q so that the iequality i 42 is shar for y c log q This is rove i Lemma 47 below Proositio 42 Let g 3 be a fixed odd iteger, ad ε > 0 be small Let ψ be a odd rimitive character of coductor m ad order k, ad y be such that m log y 4/7

14 4 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Put k = k/g, k The max 43 Re z µ g {0} l mod k z e l k y ψ=e k l π/gk = δ g taπ/gk log 2 y + θε log m + O log 2 m, where θ = 0 if m i a o-excetioal modulus, ad θ if m is excetioal We first record the followig lemma, which is a secial case of Lemma 83 of [4] Lemma 43 Let g, k ad k be as i Proositio 42 The max k Re z e l π/gk = δ g z µ g {0} k taπ/gk l mod k Proof This is Lemma 84 of [4] see also Lemma 52 below with θ = 0 I view of this lemma, our ext task is to estimate the ier sum i the left had side of 43 Sice ψ is eriodic modulo m we have 44 = y ψ=e l k a mod m ψa=e l k y a mod m I what follows we shall eed estimates of Mertes tye for sums of recirocals of rimes from secific arithmetic rogressios a modulo m that are uiform i a rage of the modulus m Results of this tye were established by Laguasco ad Zaccagii [] Lemma 44 Theorem 2 ad Corollary 3 of [] Let x 3 The, uiformly i m log x ad reduced residue classes a modulo m, we have log = log log x 6/5 φm log 2 x C m a + O, log x 3/5 x a mod m where C m a is defied i 45 below We shall refer to C m a as the Mertes costat of the residue class a modulo m Give m 2 ad a, m =, this quatity is defied by 45 C m a := K, χ χa log φm L, χ γ + logφm/m, φm χ χ 0 mod m where, for each o-ricial character χ modulo q, k χ Ks, χ := s =

15 LARGE ODD ORDER CHARACTER SUMS 5 is a absolutely coverget Dirichlet series for Res > 0, ad k χ is a comletely multilicative fuctio defied as 46 k χ := χ χ Moreover, it follows from the defiitio of k χ ad Taylor exasio that 47 k χ I order to study the asymtotic behaviour of the sum i 44, it will be crucial to have a uer boud for the average of C m a Lemma 45 Fix ε > 0, ad let m 3 The, we have { Olog C m a 2 m, if m is a o-excetioal modulus, ε log m + Olog 2 m, if m is excetioal a mod m a,m= Proof First, sice φm m/ log 2 m the C m a = φm χ χ 0 mod m χa log K, χ L, χ + O log3 m φm Let χ be a o-ricial character modulo m By 47 we have log K, χ = log k χ k χ + O x >x = log k χ + O x x Furthermore, it follows from 46 that log k χ + log χ = χ log If χ is a o-excetioal character, the Lσ + it, χ does ot vaish whe σ c logm t + 2, for some ositive costat c Therefore, takig T = m 2, σ 0 = c/4 log m ad X = exlog m 3 i Lemma 34 we obtai 48 log L, χ = log χ + O m X

16 6 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL We first cosider the case whe m is a o-excetioal modulus Usig the above estimates together with the orthogoality of characters we coclude that C m a = χa χ log log3 m + O φm φm χ χ 0 mod m X 49 = log log log3 m + O φm φm X a mod m Thus, we deduce i this case that C m a a mod m a,m= a mod m a,m= log 2 m X a mod m X m log log + Olog 3 m X Now, suose that m is a excetioal modulus, ad let χ m be the excetioal character modulo m The aroximatio 48 is valid for all o-ricial characters χ χ m modulo m Furthermore, for χ = χ m we have Siegel s boud see Theorem 4 i [4] ad hece, istead of 48 we use that log L, χ m + log X log L, χ m ε log m + O ε, χ m ε log m + Olog 2 m Thus, similarly to 49 we obtai i this case that C m a log X a mod m + ε log m φm + O log2 m φm Summig over all reduced residue classes a modulo m gives the desired boud Proositio 42 ow follows readily Proof of Proositio 42 First, ote that for each fixed l modulo k, there are exactly φm/k residue classes a modulo m such that a, m = ad ψa = e l k This follows from the simle fact that the umber of such residue classes equals the size of the kerel of ψ, ad by basic grou theory this is Z/mZ / Imψ = φm/k Thus, we deduce from 44 ad Lemma 44 that y ψ=e l k = a mod m ψa=e l k = log 2 y k log 2 y φm C ma + y am a mod m ψa=e l k C m a + y ψ=e l k log + + O log y 4/7 log + φm + O klog y 4/7

17 LARGE ODD ORDER CHARACTER SUMS 7 Summig over l modulo k, ad usig Lemma 43, we get max Re z e l z µ g {0} k l mod k y ψ=e l k π/gk = δ g taπ/gk log 2 y + θ C m a + O, a mod m a,m= for some comlex umber θ Aealig to Lemma 45 comletes the roof Let ψ be ay odd character modulo m, with eve order k I choosig characters χ of order g ad coductor q N that maximize the distace Dχ, ψ; y with y log N/0, we will eed to be able to choose the values of χ at the small rimes y Usig Eisestei s recirocity law ad the Chiese Remaider Theorem, Goldmakher [4] roved the existece of such characters Lemma 46 Proositio 93 of [4] Let g 3 be fixed, ad y be large Let {z } be a sequece of comlex umbers such that z µ g {0} for each rime There exists a ositive iteger q such that g y g q 2g y g ad a rimitive Dirichlet character χ of order g ad coductor q such that χ = z for all y with g However, i order to rove Theorem 3 we eed to fid may such characters, sice we must avoid those i the excetioal set of Theorem 25, which has size at most N ε To this ed we rove Lemma 47 Let N be large Let g 3 be fixed Let 2 y log N/0, ad ut z = z y µ g {0} πy There are N 3/4 g 2πy+2 log 2 N rimitive Dirichlet characters χ of order g ad coductor q N such that χ = z for each y such that g The secial case z = =,,, was roved by the first author i Lemma 23 of [0], but the roof there does ot aear to geeralize to all z µ g {0} πy However, we will show that oe ca combie the secial case z = with Lemma 46 i order to obtai the geeral case i Lemma 47 Proof of Lemma 47 Let S z,g N be the set of all characters χ of order g ad coductor q N such that χ = z for all y with g By Lemma 46, there exists l ad,

18 8 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL a rimitive Dirichlet character ξ of order g ad coductor l such that ξ = z for all y with g Moreover, oe has log l = y log + O g = y + o, by the rime umber theorem, ad hece l N /8 by our assumtio o y O the other had, Lemma 23 of [0] imlies that there are N 3/4 g 2πy+2 log 2 N rimitive Dirichlet characters ψ of order g ad coductor, such that = q q 2 where N 3/8 < q < q 2 < 2N 3/8 are rimes with 2 mod g, ad such that ψ = for all rimes y Now, for ay such we have l, = sice l N /8, ad hece ψ ξ is a rimitive character of order g ad coductor l N Fially observe that ψ ξ = z for each y such that g Thus we deduce that ψ ξ S z,g N for every character ψ, comletig the roof We fiish this sectio by rovig Proositio 26 ad Theorem 3 Proof of Proositio 26 Let m be a o-excetioal modulus, ad ψ be a odd rimitive character modulo m with order k For each 0 l k, suose that the maximum of Re ze l k for z µg {0} πy is attaied whe z = z l The, it follows from Lemma 47 that there are at least N rimitive characters χ of order g ad coductor q N such that y Re χψ = l mod k Re z l e l k + O g y ψ=e l k The desired result the follows from 42 ad Proositio 42 Proof of Theorem 3 Let N be sufficietly large, ad let y = log N/0 Let m be a rime umber that is also a o-excetioal modulus, such that log 3 N m 2 log 3 N Oe ca make such a choice sice it is kow that there is at most oe excetioal rime modulus betwee x ad 2x for ay x 2 see Chater 4 of [3] for a referece Let ψ be a rimitive character modulo m of order k = φm = m Note that such a character is ecessarily odd By Proositio 26, there are at least N/2 rimitive characters of order g ad coductor N /3 q N such that Dχ, ψ; y 2 π/gk = δ g log taπ/gk 2 y + Olog 2 m = δ g log 3 q + O log 5 q, sice gk k ad t/ tat = +Ot 2 Thus, sice Dχ, ψ; log q 2 = Dχ, ψ; y 2 +O, the it follows from Theorem 25 with ε = /4 that there are at least N/3 rimitive

19 LARGE ODD ORDER CHARACTER SUMS 9 characters of order g ad coductor N /3 q N such that qm Mχ φm log 2 q δg log 4 q O qlog 2 q δg log 3 q 4 log4 q O 5 Estimates for Mχψ; y, T : Proofs of Proositios 2 ad 22 5 Lower bouds for Mχψ; y, T for small twists T : Proof of Proositio 2 Let χ be a rimitive character modulo q of odd order g 3, ad ψ be a odd rimitive character of coductor m ad order k Let y exm 7/4α be a real umber, ad ut z = ex log y α Sice m log z 4/7, the it follows from Proositio 4 that for all x z we have Dχ, ψ; x 2 π/gk δ g taπ/gk 5 δ g + π2 δ g 4gk 2 log 2 x βε log m + O log 2 m log 2 x βε log m + O log 2 m sice gk 6, ad u/ tau u 2 /4 for 0 u π/6 Let t be a real umber such that t log y α First, if t log y, the sice it = + O t log we obtai 52 Dχψ, it ; y 2 = Dχ, ψ; y 2 + O t y log = Dχ, ψ; y 2 + O, ad hece the desired lower boud for Dχψ, it ; y 2 follows from 5 Thus, we ca ad will assume throughout this subsectio that t > log y Furthermore, sice t log y α = / log z, the similarly to 52 oe has Dχψ, it ; y 2 = Dχ, ψ; z 2 + Reχψ it + O z< y Therefore, i view of 5, it is eough to rove the followig result i order to deduce Proositio 2 Proositio 5 Let χ, ψ, y, z ad t be as above The we have Reχψ it log y δ g log + O log z z< y To establish this result, we will follow the argumets i Sectio 8 of [4] We shall eed the followig lemmas Lemma 52 Lemma 83 of [4] Let g 3 be odd, k 2 be eve, ad θ R Put k = k/g, k The we have max k Re z e θ l = siπ/g z µ g {0} k k taπ/gk F gk gk θ, l mod k

20 20 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL where F u := cos2π{u}/ + taπ/ si2π{u}/, ad {u} is the fractioal art of u Lemma 53 Let T > ad 3 be a ositive iteger The T F u 53 u du = π π ta log T + O, ad F u 54 du = log T + O u A /T I articular, for ay 0 < A < B we have B F u 55 u du π π ta logb/a + O, ad the costats i the O error terms are absolute Proof We first rove 53 Sice F is bouded ad -eriodic, we have T F u u du = F u du + O = F udu + O j T 0 u + j j j T 0 = π π ta log T + O The secod estimate 54 follows from observig that for u [0, ad 3 we have u 2 π u F u = + O + ta = + Ou 2 Fially, to rove 55 we cosider the three cases A < B, A < < B, ad A < B The first case follows from 53, ad the third follows from 54 uo usig the iequality taπ/ π/ Fially, i the secod case we have B A F u u du = A F u u du + B F u u du = π π ta log B log A + O, which imlies the result sice taπ/ π/ ad log A > 0 Proof of Proositio 5 Let x 0 = z, ad δ > 0 be a small arameter to be chose For each ositive iteger r R := logy/z/ log + δ, set x r := + δ r z We cosider the sum S = z< y Write θ r := so that Reχψ it = 0 r R t log xr, ad ote that if x 2π r, x r+ ] the 56 S = Reχψ it x r< x r+ it eθ r t log + δ δ t, 0 r R Reeθ r χψ + O δ x r< x r+ + O δ

21 For each 0 r R, we defie S r := Reeθ r χψ x r< x r+ xr< x r+ a mod m LARGE ODD ORDER CHARACTER SUMS 2 max z µ {0} l mod k Re ze θ r l k a mod m ψa=e l k xr< x r+ a mod m Note that m log x r 4/7 for each 0 r R Thus, by the Siegel-Walfisz theorem see Corollary 9 i [4], there is a ositive costat b such that log = x r+ x r + O x r ex b log x r, φm for all 0 r R Moreover, for x r < x r+ = + δx r, we have = log + log x r log x r = + Oδ log x r log x r x r log x r x r log x r Thus, combiig these two statemets, we get log = +Oδ δ xr< x r+ a mod m = + Oδ x r log x r xr< x r+ a mod m + Oδ δ siπ/g k taπ/gk F gk gk θ r log x r φm log x r +O ex b log z, ad uo summig over a modulo m such that ψa = e l k, of which there are φm/k as remarked i Sectio 3, we see that S r + Oδ δ max k log x Re ze θ r l + O φm ex b log z r z µ {0} k l mod k + O φm ex b log z by Lemma 52 Summig over 0 r R this yields 57 0 r R S r +Oδ δ siπ/g k taπ/gk 0 r R F tgk gk 2π log x r +O ex log y α 4 log x r sice φmr log y 3, ad z = ex log y α Recall that for 3, F u = cos2π{u}/ + taπ/ si2π{u}/ is bouded, eriodic with eriod, ad cotiuous o R sice lim u F u = F 0 Moreover, F is cotiuously differetiable o the iterval 0,, ad F u = O/ uiformly i u R \ Z It follows from these facts, together with the mea value theorem, that F a F b = O a b / for all a, b R such that a b <, where the costat i the O is absolute This shows that for all u [log x r, log x r+ we have tgk tgk F gk 2π u = F gk 2π log x r + O δ t Furthermore, we ote that log xr+ du u = + Oδ δ log x r log x r,

22 22 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Combiig these two facts, we obtai δ tgk F gk log x r 2π log x r = + Oδ log x r+ log x r = + Oδ log x r+ tgk du F gk 2π u u + O δ t log x r Summig over 0 r R, we get F tgk gk 2π δ log x r = + Oδ log x r 0 r R log y F gk log z F gk tgk log xr+ log x r 2π log x r du u du u tgk du 2π u u + O δ, sice log y log x R dt/t δ We ow estimate the itegral i the mai term above Oe ca easily check that for 3, F is a eve fuctio Makig the chage of variable v := gk t u, ad settig 2π A := gk t log z ad B := gk t log y, we get 2π 2π log y tgk du B F gk 2π u u = log z A F gk v dv v gk π ta π log gk log y + O, log z by Lemma 53 Combiig the above estimates with 56 ad 57 we obtai S + Oδ g π log y log y π si log + O δ g log + Oδ log g log z log z 2 y Choosig δ = log 2 y comletes the roof of Proositio 5 Proositio 2 follows uo combiig this result with 5 52 Estimatig Mχψ; y, T for large twists T I this subsectio, we rove the followig result which imlies Proositio 22 Proositio 54 Assume GRH Let g 3 be a fixed odd iteger Let N be large ad y log N/0 Let ψ be a odd rimitive character of coductor m such that ex 2 log 3 y m ex log y The, there exist at least N rimitive characters χ of order g ad coductor q N such that D 2 χψ, i ; y = δ g log 2 y + O log 2 m Proof We follow the roof of Proositio 5 i such a way that we achieve equality i all stes Sice the argumets here are similar to those i that roof, we omit some of the details Let z := ex log m 2 ad y z Let δ > 0 be a small arameter to be chose ad ut R := logy/z/ log + δ as before Set x 0 = z ad x r := + δ r x 0 The, as z log 2 m, it suffices to fid at least N rimitive characters χ of order g ad coductor q N such that Reχψ i 58 = δ g loglog y/ log z + O z< y

23 LARGE ODD ORDER CHARACTER SUMS 23 log xr Let θ r :=, for each 0 r R As i the roof of Proositio 5, whe 2π x r < x r+ we aroximate i by x i r, for each 0 r R Let k be the order of ψ, ad for each r let {z r,l } l µ g {0} k be chose so as to maximize the sum Re z r,l e θ r l k l mod k a mod m ψa=el/k xr< x r+ a mod m By Lemma 47 there are at least N rimitive characters χ of order g ad coductor q N such that χ = z r,l wheever x r < x r+, ψ = e l/k ad g For such characters, it follows that Let z< y = Reχψ i = Re z r,l e 0 r R l mod k S r := l mod k 0 r R θ r l k Reχψx i x r< x r+ a mod m ψa=el/k Re z r,l θ r l k a mod m ψa=el/k xr< x r+ a mod m r xr< x r+ a mod m + O δ + O g To estimate the ier sum, we use the followig asymtotic formula, which is valid uder the assumtio of GRH: This yields 0 r R xr< x r+ a mod m xr< x r+ a mod m = log = x r+ x r φm + O x /2 r log 2 x r δ + O δ + O xr 2/5 φm log x r Usig this estimate ad roceedig exactly as i the roof of Proositio 5, we obtai that siπ/g log y F gk gk S r = + O δ u 2π du + O E, k taπ/gk log z u where E δ + z 2/5 0 r R + δ 2r/5 δ + z 2/5 δ Here, ote that if we trasform the itegral as we did i the roof of Proositio 5, ie, with v := gk u the the bouds of itegratio, A := gk log z ad B := gk log y are 2π 2π 2π

24 24 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL both larger tha Thus, alyig Lemma 53, we get log y F gk gk log z u 2π du = u B A F gk v B dv = v F gk v A dv v = gk π taπ/gk logb/a + O Isertig this ito our estimate for r S r, we get S r = δ g loglog y/ log z + O 0 r R F gk v dv v + δ log 2 y + z 2 5 δ Choosig δ = log 2 y as before, ad otig that z log 2 y 4 yields 58 for y sufficietly large This comletes the roof of Proositio 54 Proositio 22 follows as well 6 Logarithmic mea values of comletely multilicative fuctios: roof of Theorem 4 The key igrediet to the roof of Theorem 4 is the followig geeralizatio of Theorem 2 of [3] Theorem 6 Let f F ad x 2 The, for ay 0 < T we have x f log x / log x H T α α dα, where ad H T α = max s A k,t α k= F + s s 2 /2 A k,t α = {s = σ + it : α σ, t kt T/2} Motgomery ad Vaugha [3] established this result for T =, ad a straightforward geeralizatio of their roof allows oe to obtai Theorem 6 for ay 0 < T For the sake of comleteess we will iclude a full sketch of the ecessary modificatios to obtai this result The oly differet treatmet occurs whe boudig the itegrals o the left had side of 6 below Lemma 62 Let 0 < α, T The we have 6 F 2 + α + it 2 α + it dt + F + α + it α + it 2 dt H T α 2 α

25 Proof First, we have F 2 + α + it α + it dt = LARGE ODD ORDER CHARACTER SUMS 25 k= kt +T/2 kt T/2 max t kt T/2 k= F + α + it α + it F + α + it α + it 2 dt 2 kt +T/2 kt T/2 F + α + it F + α + it To boud the itegral o the right had side of this iequality, we aeal to a result of Motgomery see Lemma 6 of [7] which states that if a s ad b s are two Dirichlet series which are absolutely coverget for Res > ad satisfy a b for all, the we have 62 u u 2 a σ+it = u dt 3 u 2 b σ+it = for ay real umbers u 0 ad σ > This imlies that kt +T/2 F α + it T/2 Λf F + α + it dt = dt +α+ikt +it kt T/2 Hece, we deduce that To comlete the roof, ote that 2 F + α + it α + it 2 dt T/2 = T/2 T/2 α + it 2 dt F 2 + α + it α + it dt H T α 2 α max t kt T/2 k= F + α + it α + it dt, T/2 T/2 α 2 + t 2 dt α 2 kt +T/2 kt T/2 ζ + α + it ζ + α + it 2 2 dt dt α + it dt H T α 2 2 α Proof of Theorem 6 Let Sx = x f From the Euler roduct, F 2 > 0, so H T α Thus, it is eough to rove the statemet for x x 0, where x 0 is a suitably large costat Moreover, observe that / log x H T αα dα is strictly icreasig as a fuctio of x, ad Sx log x is strictly icreasig for x [, +, for all Hece it is eough to rove the result for x B where B = {x x 0 : Sy log y < Sx log x for all y < x}

26 26 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Motgomery ad Vaugha roved that for x B we have see equatios 7 ad 8 of [3] x Su Sx log x u du+ f x log x log log + f x log x log2 e x Itegratig the first itegral by arts, we get 63 where Ju := u e x e Su u Jx du log x + x St log t u dt log u /2 t e e Ju ulog u 2 du, x St 2 log t 2 /2 dt, t by the Cauchy-Schwarz iequality Usig Parseval s Theorem, Motgomery ad Vaugha roved that see equatio 4 of [3] u St 2 log t 2 dt F 2 + β + it 2 t β + it dt + F + β + it β + it 2 dt, e e where β = 2/ log u Aealig to Lemma 62 ad makig the chage of variable α = / log u i the itegral of the right had side of 63 we deduce that x Su 2 64 u du H H T 2α T + dα log x α / log x Sice H T α is decreasig as a fuctio of α, we have 2 2/ log x H T α 65 H T log x α dα Combiig 64 ad 65 we get x x e / log x Su u du / log x / log x H T α α dα H T α α dα Furthermore, Motgomery ad Vaugha roved that see ages of [3] f x log log F 2 /2 + β + it β β + it dt ad x f x log2 β F + β + it β + it 2 2 dt /2, where β = 2/ log x Combiig these bouds with Lemma 62 ad equatio 65 comletes the roof

27 LARGE ODD ORDER CHARACTER SUMS 27 I order to derive Theorem 4 from Theorem 6, we eed to boud H T α, ad hece to boud F + s for Res α Teebaum see Sectio III4 of [7] roved that for all y, T 2, ad / log y α, we have 66 max t T F + α + it log y ex Mf; y, T However, this boud does ot hold for all T > 0 ad / log y α Ideed, takig f to be the Möbius fuctio µ, α = /2, y large ad T = / log y shows that max t T F + α + it ζ3/2, while Mf; y, T = + Re it mi t / log y y = 2 y + O = 2 log log y + O, ad hece the right side of 66 is /log y Nevertheless, usig Teebaum s ideas, we show that 66 is valid wheever T α Lemma 63 Let y 2 ad f F such that f = 0 for > y Let F s be its corresodig Dirichlet series The, for all real umbers 0 < α ad T α we have Proof Note that max F + α + it log y ex Mf; y, T t T 67 Mf; y, T = log 2 y max t T Re y f + O +it We first remark that the result is trivial if α / log y, sice i this case we have log F + α + it = Re y f +α+it + O = Re y f + O, +it which follows from the fact that α α log Now, suose that α / log y ad ut A = ex/α The we have log F +α+it = Re y f +α+it +O = Re A f +α+it +O = Re A f +O, +it sice >A α by the rime umber theorem Furthermore, for ay β α/2 we have f = f + O, +it+β +it A A ad hece max F + α + it max ex t T t T α/2 Re A f +it

28 28 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL Now, let t T α/2 be a real umber The, we have t+α/2 f Re du = Re f iα/2 iα/2 t α/2 +iu +it i log y y = α Re f + O α + +it log A >A Sice >A log α by the rime umber theorem, we deduce that Re f = t+α/2 f Re du + O max +it α A t α/2 Re f + O +iu t T +it y y Aealig to 67 comletes the roof We fiish this sectio by rovig a slightly stroger form of Theorem 4, which we shall eed to rove Theorems ad 2 Oe ca also show that the followig result follows from Theorem 4, so it is i fact equivalet to it Theorem 64 Let f F ad x, y 2 be real umbers The, for ay real umber 0 < T we have x Sy f log y ex Mf; y, T + T, where the imlicit costat is absolute, ad Sy is the set of y-friable umbers Proof First, observe that the result is trivial if T / log x, sice we have i this case f log x T x x Sy Now assume that / log x < T Let g be the comletely multilicative fuctio such that g = f for y ad g = 0 otherwise, ad let G be its corresodig Dirichlet series The, it follows from Theorem 6 that 68 where x Sy f H T α = = x g log x max s A k,t α k= / log x G + s s H T α α dα, 2 /2 First, observe that if t kt T/2 ad k 0 the t k T Moreover, uiformly for all t R, we have 69 G + σ + it ζ + σ σ

29 LARGE ODD ORDER CHARACTER SUMS 29 We will first boud H T α whe α > T Usig 69 we obtai i this case 60 α 2 H T α 2 max t kt T/2 k= α 2 + t 2 k >α/t k 2 T 2 + k α/t α 2 αt Now, suose that 0 < α T To boud H T α i this case, we first use 69 for k This gives H T α 2 α 2 k k 2 T 2 + α 2 Furthermore, by 69 ad Lemma 63 we have max s A 0,T α G + s max t T σ T max G + s A 0,T α s 2 αt + 2 α 2 max G + s A 0,T α s 2 G + σ + it + max G + σ + it t T α σ T T + log y ex Mg; y, T Sice Mg; y, T = Mf; y, T we deduce that for 0 < α T we have 6 H T α 2 log y2 + ex 2Mf; y, T αt 2 α 2 Usig 60 whe T < α ad 6 whe / log x α T we get / log x H T α α dα T log x T T + log y ex Mf; y, T / log x + log xlog y ex Mf; y, T Isertig this boud i 68 yields the result α dα + 2 T /2 T dα α5/2 7 Proofs of Theorems 23, ad 2 To rove Theorem 23, the geeral strategy we use is that of [9] with the refiemets from [4], ad it will be clear where we shall make use of Theorem 64 We will cosider the coditioal o GRH ad ucoditioal results simultaeously, settig y := log 2 q if we are assumig GRH, ad settig y := q otherwise We recall here that y = Q i the ucoditioal case, ad y = Q 2 o GRH, so that i all cases we have log y log Q Whe χ is rimitive ad α R, we have q χ eα = q Sy χ eα + O;

30 30 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL o GRH, this follows from 25, ad ucoditioally this statemet is trivial Isertig this estimate i Pólya s Fourier exasio 2 gives Mχ χ q max eα + α [0,] q Sy Therefore, to rove Theorem 23 it suffices to show that for all α [0, ] we have 7 q Sy χ e α m χ ξ φm log Qe Mχξ;Q,log Q 7/ +log Q 9 +o Let α [0, ] ad R := log Q 5 By Dirichlet s theorem o Diohatie aroximatio, there exists a ratioal aroximatio α b/r /rr, with r R ad b, r = Let M := log Q 4/ We shall distiguish betwee two cases If r M, we say that α lies o a major arc, ad if M < r R we say that α lies o a mior arc I the latter case, we shall use Corollary 22 of [4], which is a cosequece of the work of Motgomery ad Vaugha [5] Ideed, this shows that q Sy χ log M5/2 e α log y + log R + log 2 y log Q 9 M +o We ow hadle the more difficult case of α lyig o a major arc First, it follows from Lemma 4 of [4] which is a refiemet of Lemma 62 of [9] that for N := mi{q, rα b }, we have 72 q Sy χ eα = N Sy = N Sy χ b log R 3/2 e + O log y 2 + log R + log r 2 y R χ b e + O log r 2 Q We first assume that b 0 I this case we ca use a idetity of Graville ad Soudararaja see Proositio 23 of [4] which asserts that 73 N Sy = χ b e r χ ψ d r d Sy χd d φr/d ψ mod r/d τψψb N/d Sy χψ

31 LARGE ODD ORDER CHARACTER SUMS 3 To boud the ier sum above, we aeal to Theorem 64 with T = log Q 7/ This imlies that N/d Sy χψ log y ex Mχψ; y, log Q 7/ + log Q 7/ Moreover, i the coditioal case y = Q 2, ad thus we have Therefore, we get 74 N/d Sy Mχψ; y, log Q 7/ Mχψ; Q, log Q 7/ + O χψ log Q ex Mχψ; Q, log Q 7/ + log Q 7/ We ow order the rimitive characters ψ mod l for l M icludig the trivial character ψ which equals for all itegers as {ψ k } k, where Mχψ k ; Q, log Q 7/ Mχψ k+ ; Q, log Q 7/, for all k Note that ψ = ξ, i the otatio of Theorem 23 Furthermore, by a slight variatio of Lemma 3 of [] we have M χψ k ; Q, log Q 7/ log 2 Q + O log2 Q k Therefore, if ψ mod l is iduced by ψ k, the 75 M χψ; Q, log Q 7/ M χψ k ; Q, log Q 7/ + O l + o log 2 Q, k sice l / log 2 l log 3 Q Isertig this boud i 74, we deduce that the cotributio of all characters ψ that are iduced by some ψ k with k 3 to 73 is log Q 7/ τψ log Q r 7/ dφr/d d log 3/2 Q9/, d r ψ mod r/d d r sice / 3 < 7/, τψ r/d, ad r log Q 4/ Moreover, observe that there is at most oe character ψ mod r/d such that ψ is iduced by ψ 2 Usig 75, we deduce that the cotributio of these characters to 73 is log Q / 2+o d r r/d d φr/d log 2+o Q/ log r log Q / 2+o Thus, it ow remais to estimate the cotributio of the characters ψ mod r/d that are iduced by ξ, recallig that ξ has coductor m If m r, there are o such characters

32 32 YOUNESS LAMZOURI AND ALEXANDER P MANGEREL ψ ad the theorem follows i this case If m r ad ψ mod r/d is iduced by ξ, the we must have d r/m Furthermore, by Lemma 4 of [9] we have r r τψ = µ ξ τξ dm dm Therefore, the cotributio of these characters to 73 is 76 χ ξ ξbτξ d r/m d Sy χd d Furthermore, it follows from Lemma 44 of [9] that N/d,r/d= Sy χξ = N,r/d= Sy = r d d r/m r/dm,m= χξ χξ r r φr/d µ ξ dm dm N/d,r/d= Sy + Olog d N Sy χξ χξ + O log 2 Q 2 Thus, i view of Theorem 64, we deduce that 76 is m χ ξ log Qe Mχξ;Q,log Q 7/ + log Q 7/ 77 r dφr/d µ2 + dm d r/m r/dm,m= r dm Fially, by a chage of variables a = r/md, we obtai r dφr/d µ2 + = m dm rφm r dm a r/m a,m= φm r/m a φa µ2 a a r/m φm, sice 2/ for all rimes 3 Combiig this boud with 77, it follows that the cotributio of the characters ψ that are iduced by ξ to 73 is m χ ξ φm log Qe Mχξ;Q,log Q 7/ + log Q 7/ It thus remais to cosider whe b = 0, ad hece r = First, if ξ is idetically so m =, the a trivial alicatio of Theorem 64 shows that i this case χ m χ φm log Qe Mχ;Q,log Q 7/ + log Q 7/ N Sy

33 LARGE ODD ORDER CHARACTER SUMS 33 O the other had, if ξ is ot the trivial character, the it follows from 75 that Mχ; Q, log Q 7/ + o log 2 Q, 2 ad hece by Theorem 64 we get N Sy χ log 2+o Q/, which comletes the roof of 7 Theorem 23 follows as well We ed this sectio by deducig Theorems ad 2 from Theorem 23 ad Proositio 2 We shall rove both results simultaeously, by settig Q := log q o GRH ad Q := q ucoditioally Proof of Theorems ad 2 Let ξ be the character of coductor m log Q 4/ that miimizes M χψ; Q, log Q 7/ If ξ is eve, the it follows from Theorem 23 that Mχ qlog Q 9/+o, which trivially imlies the result i this case sice δ g > 9/, for all g 3 Now, suose that ξ is odd ad let k be its order We also let β = if m is a excetioal modulus, ad β = 0 otherwise The, combiig Theorem 23 ad Proositio 2 with α = 7/ we obtai 78 qm Mχ φm log Q δg ex c δ g qlog Q δg ex 2 βε log gk 2 2 Q + βε log m + O log 2 m log m c δ g g 2 m 2 log 2 Q + c 2 log 2 m, for some ositive costats c, c 2, sice φm m/ log 2 m Oe ca easily check that the exressio iside the exoetial is maximal whe m log 2 Q, ad its maximum equals log 3 Q + O log 4 Q 4 βε 2 Isertig this estimate i 78 comletes the roof Refereces [] A Balog, A Graville ad K Soudararaja Multilicative fuctios i arithmetic rogressios A Math Qu , o, 3 30 [2] E Bombieri, O the Large Sieve Mathematika, 2 965, [3] H Daveort, Multilicative umber theory Third editio Revised ad with a reface by Hugh L Motgomery Graduate Texts i Mathematics, 74 Sriger-Verlag, New York, 2000 xiv+77 [4] L Goldmakher, Multilicative mimicry ad imrovemets of the Pólya-Viogradov iequality Algebra Number Theory 6 202, o, [5] L Goldmakher ad Y Lamzouri, Lower bouds o odd order character sums, It Math Res Not , o 2,