Large odd order character sums and improvements of the Pólya-Vinogradov inequality
aa r X i v : . [ m a t h . N T ] J a n LARGE ODD ORDER CHARACTER SUMS AND IMPROVEMENTSOF THE P ´OLYA-VINOGRADOV INEQUALITY
YOUNESS LAMZOURI AND ALEXANDER P. MANGEREL
Abstract.
For a primitive Dirichlet character χ modulo q , we define M ( χ ) =max t | P n ≤ t χ ( n ) | . In this paper, we study this quantity for characters of a fixedodd order g ≥
3. Our main result provides a further improvement of the classicalP´olya-Vinogradov inequality in this case. More specifically, we show that for anysuch character χ we have M ( χ ) ≪ ε √ q (log q ) − δ g (log log q ) − / ε , where δ g := 1 − gπ sin( π/g ). This improves upon the works of Granville and Soundarara-jan and of Goldmakher. Furthermore, assuming the Generalized Riemann hypothesis(GRH) we prove that M ( χ ) ≪ √ q (log q ) − δ g (log q ) − (log q ) O (1) , where log j is the j -th iterated logarithm. We also show unconditionally that thisbound is best possible (up to a power of log q ). One of the key ingredients in theproof of the upper bounds is a new Hal´asz-type inequality for logarithmic mean valuesof completely multiplicative functions, which might be of independent interest. Introduction
The study of Dirichlet characters and their sums has been a central topic in analyticnumber theory for a long time. Let q ≥ χ be a non-principal Dirichlet charactermodulo q . An important quantity associated to χ is M ( χ ) := max t ≤ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ t χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The best-known upper bound for M ( χ ), obtained independently by P´olya and Vino-gradov in 1918, reads(1.1) M ( χ ) ≪ √ q log q. Though one can establish this inequality using only basic Fourier analysis, improvingon it has proved to be a difficult problem, and resisted substantial progress for severaldecades. Conditionally on the Generalized Riemann Hypothesis (GRH), Montgomeryand Vaughan [15] showed in 1977 that(1.2) M ( χ ) ≪ √ q log log q. Mathematics Subject Classification.
Primary 11L40.The first author is partially supported by a Discovery Grant from the Natural Sciences and Engi-neering Research Council of Canada.
This bound is best possible in view of an old result of Paley [16] that there exists aninfinite family of primitive quadratic characters χ mod q such that(1.3) M ( χ ) ≫ √ q log log q. Assuming GRH, Granville and Soundararajan [9] extended Paley’s result to charactersof a fixed even order 2 k ≥
4. The assumption of GRH was later removed by Goldmakherand Lamzouri [6], who obtained this result unconditionally, and subsequently Lamzouri[10] obtained the optimal implicit constant in (1.3) for even order characters.The situation is quite different for odd order characters. In this case, Granville andSoundararajan [9] proved the remarkable result that both the P´olya-Vinogradov andthe Montgomery-Vaughan bounds can be improved. More specifically, if g ≥ χ is a primitive character of order g and conductor q then they showedthat(1.4) M ( χ ) ≪ √ q (log Q ) − δg + o (1) , where δ g := 1 − gπ sin( π/g ) and(1.5) Q := ( q unconditionally , log q on GRH . By refining their method, Goldmakher [4] was able to obtain the improved bound(1.6) M ( χ ) ≪ √ q (log Q ) − δ g + o (1) . Our first result gives a further improvement of the P´olya-Vinogradov inequality for M ( χ ) when χ has odd order g ≥
3. Here and throughout, we write log k x = log(log k − x )to denote the k th iterated logarithm, where log x = log x . Theorem 1.1.
Let g ≥ be a fixed odd integer, and let ε > be small. Then, for anyprimitive Dirichlet character χ of order g and conductor q we have M ( χ ) ≪ ε √ q (log q ) − δ g (log log q ) − + ε . The occurrence of ε in the exponent of log log q in the upper bound is a consequenceof the possible existence of Siegel zeros. In particular, if Siegel zeros do not exist thenthe (log log q ) ε term can be replaced by (log q ) O (1) . Assuming GRH, and using results of Granville and Soundararajan (see Theorem2.4 below), Goldmakher [4] also showed that the conditional bound in (1.6) is bestpossible. More precisely, for every ε > g ≥
3, he proved the existenceof an infinite family of primitive characters χ mod q of order g such that(1.7) M ( χ ) ≫ ε √ q (log log q ) − δ g − ε , conditionally on the GRH. By modifying the argument of Granville and Soundarara-jan and using ideas of Paley [16], Goldmakher and Lamzouri [5] proved this resultunconditionally. ARGE ODD ORDER CHARACTER SUMS 3
It is natural to ask to what degree of precision we can determine the exact order ofmagnitude of the maximal values of M ( χ ) when χ has odd order g ≥
3; in particular,can we determine the optimal (log log q ) o (1) contributions in the conditional part of(1.6), and in (1.7). We make progress in this direction by showing that this term can bereplaced by (log q ) − (log q ) O (1) in both (1.6) and (1.7). This allows us to conditionallydetermine the maximal values of M ( χ ), up to a power of log q. Theorem 1.2.
Assume GRH. Let g ≥ be a fixed odd integer. Then for any primitiveDirichlet character χ of order g and conductor q we have (1.8) M ( χ ) ≪ √ q (log q ) − δ g (log q ) − (log q ) O (1) . Theorem 1.3.
Let g ≥ be a fixed odd integer. There are arbitrarily large q andprimitive Dirichlet characters χ modulo q of order g such that (1.9) M ( χ ) ≫ √ q (log q ) − δ g (log q ) − (log q ) O (1) . To obtain Theorem 1.3, our argument relates M ( χ ) to the values of certain asso-ciated Dirichlet L -functions at 1, and uses zero-density results and ideas from [10] toconstruct characters χ for which these values are large. We shall discuss the differentingredients in the proofs of Theorems 1.1, 1.2 and 1.3 in detail in the next section.Recent progress on character sums was made possible by Granville and Soundarara-jan’s discovery of a hidden structure among the characters χ having large M ( χ ). Inparticular, they show that M ( χ ) is large only when χ pretends to be a character of smallconductor and opposite parity. To define this notion of pretentiousness , we need somenotation. Here and throughout we denote by F the class of completely multiplicativefunctions f such that | f ( n ) | ≤ n . For f, g ∈ F we define D ( f, g ; y ) := X p ≤ y − Re( f ( p ) g ( p )) p ! , which turns out to be a pseudo-metric on F (see [9]). We say that f pretends to be g (up to y ) if there is a constant 0 ≤ δ < D ( f, g ; y ) ≤ δ log log y .One of the key ingredients in the proof of (1.4) is the following bound for logarithmicmean values of functions f ∈ F in terms of D ( f, x ) (see Lemma 4.3 of [9])(1.10) X n ≤ x f ( n ) n ≪ (log x ) exp (cid:18) − D ( f, x ) (cid:19) . Note that the factor 1 / δ g / δ g in (1.6) by re-placing (1.10) by a Hal´asz-type inequality for logarithmic mean values of multiplicativefunctions due to Montgomery and Vaughan [13]. Combining Theorem 2 of [13] with YOUNESS LAMZOURI AND ALEXANDER P. MANGEREL refinements of Tenenbaum (see Chapter III.4 of [17]) he deduced that (see Theorem2.4 in [4])(1.11) X n ≤ x f ( n ) n ≪ (log x ) exp (cid:0) − M ( f ; x, T ) (cid:1) + 1 √ T , for all f ∈ F and T ≥
1, where M ( f ; x, T ) := min | t |≤ T D ( f, n it ; x ) . Motivated by our investigation of character sums, we are interested in characterizingthe functions f ∈ F that have a large logarithmic mean, in the sense that(1.12) X n ≤ x f ( n ) n ≫ (log x ) α , for some 0 < α ≤
1. Taking T = 1 in (1.11) shows that this happens only when f pretends to be n it for some | t | ≤
1. However, observe that X n ≤ x n it n = x it − it it + O (1) ≍ min (cid:18) | t | , log x (cid:19) , and hence f ( n ) = n it satisfies (1.12) only when | t | ≪ (log x ) − α . By refining the ideasof Montgomery and Vaughan [13] and Tenenbaum [17], we prove the following result,which shows that this is essentially the only case. Theorem 1.4.
Let f ∈ F and x ≥ . Then, for any real number < T ≤ we have X n ≤ x f ( n ) n ≪ (log x ) exp (cid:0) − M ( f ; x, T ) (cid:1) + 1 T , where the implicit constant is absolute.
Taking T = c (log x ) − α in this result (where c > f ∈ F satisfies (1.12), then f pretends to be n it for some | t | ≪ (log x ) − α .Theorem 1.4 will be one of the key ingredients in obtaining our superior bounds for M ( χ ) in Theorems 1.1 and 1.2.2. Detailed statement of results
To explain the key ideas in the proofs of Theorems 1.1, 1.2 and 1.3, we shall firstsketch the argument of Granville and Soundararajan [9]. Their starting point is P´olya’sFourier expansion (see section 9.4 of [14]) for the character sum P n ≤ t χ ( n ), which reads(2.1) X n ≤ t χ ( n ) = τ ( χ )2 πi X ≤| n |≤ N χ ( n ) n (cid:18) − e (cid:18) − ntq (cid:19)(cid:19) + O (cid:18) q log qN (cid:19) , where χ is a primitive character modulo q , e ( x ) := e πix and τ ( χ ) is the Gauss sum τ ( χ ) := q X n =1 χ ( n ) e (cid:16) nq (cid:17) . ARGE ODD ORDER CHARACTER SUMS 5
Note that | τ ( χ ) | = √ q whenever χ is primitive.Thus, in order to estimate M ( χ ), one needs to understand the size of the exponentialsum(2.2) X ≤| n |≤ q χ ( n ) n e ( nθ ) , for θ ∈ [0 , θ belongsto a minor arc , i.e., θ can only be well-approximated by rationals with large denomi-nators (compared to q ). This leaves the more difficult case of θ lying in a major arc .In this case, θ can be well-approximated by some rational b/r with suitably small r (compared to q ). Granville and Soundararajan showed that in this case there is somelarge N (depending on θ , b , r and q ) such that we can approximate the sum (2.2) by X ≤| n |≤ N χ ( n ) n e ( bn/r ) = X a mod r e ( ab/r ) X ≤| n |≤ Nn ≡ a mod r χ ( n ) n = 1 φ ( r ) X ψ mod r X a mod r ψ ( a ) e ( ab/r ) ! X ≤| n |≤ N χ ( n ) ψ ( n ) n . The bracketed term, a Gauss sum, is well understood; in particular it has norm ≤ √ r ∗ ,where r ∗ is the conductor of ψ (see e.g., Theorem 9.7 of [14]). Thus, what remains tobe determined in order to bound M ( χ ), is an upper bound for the sum(2.3) X ≤| n |≤ N χ ( n ) ψ ( n ) n for each character ψ modulo r . Furthermore, observe that if χ and ψ have the sameparity then this sum is exactly 0; hence, we only need to consider the case when χ and ψ have opposite parities.Granville and Soundararajan’s breakthrough stems from their discovery of a “re-pulsion” phenomenon between characters χ of odd order (which are necessarily of evenparity), and characters ψ of odd parity and small conductor. A consequence of this phe-nomenon is that the sum (2.3) is small, allowing them to improve the P´olya-Vinogradovinequality in this case. More specifically, they show that if χ is a primitive character ofodd order g ≥ ψ is an odd primitive character of conductor m ≤ (log y ) A then(2.4) D ( χ, ψ ; y ) ≥ ( δ g + o (1)) log log y (see Lemma 3.2 of [9]). Inserting this bound in (1.10) allows them to bound the sum(2.3), from which they deduce the unconditional case of (1.4). The proof of the condi-tional part of (1.4) (when Q = log q ) proceeds along the same lines, but uses an addi-tional ingredient, namely the following approximation for the sum (2.2) (see Proposition YOUNESS LAMZOURI AND ALEXANDER P. MANGEREL X n ≤ q χ ( n ) n e ( nθ ) = X n ≤ qn ∈S ( y ) χ ( n ) n e ( nθ ) + O (cid:0) y − / (log q ) (cid:1) . Here, S ( y ) is the set of y - friable integers (also known as y - smooth integers), i.e., theset of positive integers n whose prime factors are all less than or equal to y .In [4], Goldmakher showed that the bound (2.4) is best possible. Furthermore, inorder to obtain the exponent δ g in (1.6), he used the inequality (1.11) to bound thesum (2.3) in terms of M ( χψ ; y, T ). However, to ensure that this argument works, oneneeds to show that the lower bound (2.4) still persists if we twist χψ by Archimedeancharacters n it for | t | ≤ T . By a careful analysis of M ( χψ ; y, T ), Goldmakher (seeTheorem 2.10 of [4]) proved that (under the same assumptions as (2.4))(2.6) M ( χψ ; y, (log y ) ) ≥ ( δ g + o (1)) log log y. Thus, by combining this bound with (1.11) and following closely the argument in [9],he was able to obtain (1.6).In order to improve these results and establish Theorems 1.1 and 1.2, the first stepis to obtain more precise estimates for the quantity M ( χψ ; y, T ). We discover thatthere is a substantial difference between the sizes of M ( χψ ; y, T ) and M ( χψ ; y, T ) if T is small and T is large (a result that may be surprising in view of (2.4) and (2.6)).In fact, we prove that there is a large secondary term of size (log y ) /k (where k isthe order of ψ ) that appears in the estimate of M ( χψ ; y, T ) when T ≤ (log y ) − c (forsome constant c > T ≥ Proposition 2.1.
Let g ≥ be a fixed odd integer, α ∈ (0 , , and ε > be small.Let χ be a primitive character of order g and conductor q . Let ψ be an odd primitivecharacter modulo m , with m ≤ (log y ) α/ . Put k ∗ := k/ ( k, g ) . Then we have (2.7) M ( χψ ; y, (log y ) − α ) ≥ (cid:18) δ g + απ (1 − δ g )4( gk ∗ ) (cid:19) log y − βε log m + O α (log m ) , where β = 1 if m is an exceptional modulus and β = 0 otherwise. Proposition 2.2.
Assume GRH. Let g ≥ be a fixed odd integer. Let N be large,and y ≤ (log N ) / . Let ψ be an odd primitive character of conductor m such that exp (cid:0) p log y (cid:1) ≤ m ≤ exp (cid:0) √ log y (cid:1) . Then, there exist at least √ N primitive characters χ of order g and conductor q ≤ N , such that for all T ≥ we have M ( χψ ; y, T ) ≤ δ g log y + O (log m ) . The secondary term of size ≍ (log y ) /k in the right hand side of (2.7) is responsiblefor the additional saving of (log Q ) − / (where Q is defined in (1.5)) in Theorems 1.1and 1.2; clearly, it does not appear in Proposition 2.2, even in the range m ≪ (log y ) − ε ARGE ODD ORDER CHARACTER SUMS 7 in which this secondary term is large. Note that when m is an exceptional modulus (seethe precise definition in (4.1) below), there is an additional term that appears whenestimating M ( χψ ; y, (log y ) − α ) that has size log L (1 , χ m ), where χ m is the exceptionalcharacter modulo m . In this case, the extra term ε log m on the right hand side of (2.7)is due to Siegel’s bound L (1 , χ m ) ≫ ε m − ε .To complete the proofs of Theorems 1.1 and 1.2, we shall use our Theorem 1.4to bound the sum (2.3), where we might choose T = (log y ) − α to take advantage ofProposition 2.1. Note that in view of Proposition 2.2, one loses the additional savingof (log Q ) − / in Theorems 1.1 and 1.2 if one simply uses (1.11) with T = (log y ) ,as in [4]. By using Theorem 1.4 and following the ideas in [9], we prove the followingresult, which is a refinement of Theorem 2.9 in [4]. This together with Proposition 2.1implies both Theorems 1.1 and 1.2. Theorem 2.3.
Let χ be a primitive character modulo q , and let Q be as in (1.5) . Of allprimitive characters with conductor below (log Q ) / , let ξ modulo m be that characterfor which M (cid:0) χξ ; Q, (log Q ) − / (cid:1) is a minimum. Then we have M ( χ ) ≪ (cid:16) − χ ( − ξ ( − (cid:17) √ qmφ ( m ) (log Q ) exp (cid:16) −M (cid:16) χξ ; Q, (log Q ) − (cid:17)(cid:17) + √ q (log Q ) + o (1) . Note that δ g is decreasing as a function of g , so 1 − δ g ≥ − δ ≈ . > / g ≥
3. Therefore, when χ is a primitive character of odd order g ≥ q , we get the better bound M ( χ ) ≪ √ q (log Q ) + o (1) , unless ξ is odd and M (cid:16) χξ ; Q, (log Q ) − (cid:17) is small.We next discuss the ideas that go into the proof of Theorem 1.3. To obtain (1.7)under GRH, Goldmakher [4] used the following result from [9], which relates M ( χ ) tothe distance between χ and any primitive character ψ with small conductor and parityopposite to that of χ . Theorem 2.4 (Theorem 2.5 of [9]) . Assume GRH. Let χ mod q and ψ mod m beprimitive characters such that χ ( −
1) = − ψ ( − . Then we have M ( χ ) + √ qmφ ( m ) log q ≫ √ qmφ ( m ) (log q ) exp (cid:0) − D ( χ, ψ ; log q ) (cid:1) . Thus, it only remains to produce characters χ and ψ which satisfy the assumptions ofTheorem 2.4, and for which the lower bound (2.4) is attained when y = log q . Usingthe Eisenstein reciprocity law, Goldmakher (see Proposition 9.3 of [4]) proved that forany ε >
0, there exists an odd primitive character ψ modulo m ≪ ε
1, and an infinitefamily of primitive characters χ mod q of order g such that(2.8) D ( χ, ψ ; log q ) ≤ ( δ g + ε ) log q. To remove the assumption of GRH, Goldmakher and Lamzouri [5] (see Theorem 1 of[5]) used ideas of Paley [16] to obtain a weaker version of Theorem 2.4 unconditionally.
YOUNESS LAMZOURI AND ALEXANDER P. MANGEREL
Namely, they showed that if χ is odd and ψ is even then M ( χ ) + √ q ≫ √ qmφ ( m ) (cid:18) log q log q (cid:19) exp (cid:0) − D ( χ, ψ ; log q ) (cid:1) . Although this bound is enough to obtain (1.7) unconditionally in view of (2.8), it isnot sufficient to yield the precise estimate in Theorem 1.3, due to the loss of a factorof log q over Theorem 2.4.Using a completely different method, based on zero density estimates for Dirichlet L -functions, we recover the original bound of Granville and Soundararajan uncondi-tionally for all characters χ modulo q with q ≤ N , except for a small exceptional setof cardinality ≪ N ε . Our argument also gives a simple proof of Theorem 2.4, whichexploits the natural properties of the values of Dirichlet L -functions at 1, and avoidsthe difficult study of exponential sums with multiplicative functions (see Section 6 of[9]). Note that the statement of Theorem 2.4 trivially holds when m > log q , since D ( χ, ψ ; log q ) ≪ log q . We thus only need to consider the case m ≤ log q . Theorem 2.5.
Let ε > and let N be large. Let m ≤ log N be a positive integerand let ψ be a primitive character modulo m . Then, for all but at most N ε primitivecharacters χ modulo q with q ≤ N and such that χ ( −
1) = − ψ ( − we have (2.9) M ( χ ) + √ q ≫ ε √ qmφ ( m ) (log q ) exp (cid:0) − D ( χ, ψ ; log q ) (cid:1) . Moreover, if we assume GRH, then (2.9) is valid for all primitive characters χ modulo q with q ≤ N , and the implicit constant in (2.9) is absolute. To complete the proof of Theorem 1.3, we thus need to refine the estimate (2.8), andthis can be achieved using the same ideas as in the proof of Proposition 2.1. However,Goldmakher’s proof of (2.8) only produces an infinite sequence of primitive characters χ , and this is not enough to use in Theorem 2.5, due to the possible existence of anexceptional set of characters for which (2.9) does not hold. To overcome this difficulty,we use the results of [10] to prove the existence of many primitive characters χ of order g and conductor q ≤ N such that when y ≪ log N , D ( χ, ψ ; y ) is maximal. Proposition 2.6.
Let g ≥ be a fixed odd integer. Let N be large and y ≤ (log N ) / be a real number. Let m be a non-exceptional modulus such that m ≤ (log y ) / , andlet ψ be an odd primitive character of conductor m . Let k be the order of ψ and put k ∗ = k/ ( g, k ) . Then, there exist at least √ N primitive characters χ of order g andconductor q ≤ N such that (2.10) D ( χ, ψ ; y ) = (cid:18) − (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) (cid:19) log y + O (log m ) . ARGE ODD ORDER CHARACTER SUMS 9 A lower bound for M ( χ ) : Proof of Theorem 2.5 The main ingredient in the proof of Theorem 2.4 of [9] is the approximation (2.5),which is valid under the assumption of GRH. To avoid this assumption, we shall insteadrelate M ( χ ) to the values of certain Dirichlet L -functions at s = 1, and then use theclassical zero-density estimates for these L -functions. Proposition 3.1.
Let q be large and m ≤ q/ (log q ) . Let χ mod q and ψ mod m beprimitive characters such that ψ ( −
1) = − χ ( − . Then we have M ( χ ) + √ q ≫ √ qmφ ( m ) · (cid:12)(cid:12) L (cid:0) , χψ (cid:1)(cid:12)(cid:12) . We first need the following lemma.
Lemma 3.2.
Let q be large and m ≤ q/ (log q ) . Let χ be a character modulo q and ψ be a character modulo m such that χψ is non-principal. Then L (1 , χψ ) = X n ≤ q χ ( n ) ψ ( n ) n + O (1) . Proof.
Note that χψ is a non-principal character of conductor at most qm ≤ ( q/ log q ) .Therefore, using partial summation and the P´olya-Vinogradov inequality we obtain X q Taking N = q in (2.1) gives M ( χ ) + log q ≫ √ q · max θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤| n |≤ q χ ( n ) n (1 − e ( nθ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Moreover, we observe that X b mod m ψ ( b ) X ≤| n |≤ q χ ( n ) n (cid:18) − e (cid:18) nbm (cid:19)(cid:19) = − X ≤| n |≤ q χ ( n ) n X b mod m ψ ( b ) e (cid:18) nbm (cid:19) = − τ ( ψ ) X ≤| n |≤ q χ ( n ) ψ ( n ) n , which follows from the identity X b mod m ψ ( b ) e (cid:18) nbm (cid:19) = ψ ( n ) τ ( ψ ) . Since χ and ψ are primitive and m ≤ q/ (log q ) then χψ is non-principal. Therefore,by Lemma 3.2 together with the fact that χψ ( − 1) = − X ≤| n |≤ q χ ( n ) ψ ( n ) n = 2 X ≤ n ≤ q χ ( n ) ψ ( n ) n = 2 L (1 , χψ ) + O (1) . The result follows upon noting that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X b mod m ψ ( b ) X ≤| n |≤ q χ ( n ) n (cid:18) − e (cid:18) nbm (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ φ ( m ) · max θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤| n |≤ q χ ( n ) n (1 − e ( nθ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and that | τ ( ψ ) | = √ m by the primitivity of ψ . (cid:3) In order to complete the proof of Theorem 2.5, we need to approximate L (1 , χψ )by a short truncation of its Euler product. Using zero density estimates, we prove thatthis is possible for almost all primitive characters χ . Proposition 3.3. Fix < ε < and let A = 100 /ε . Let N be large and m ≤ log N .Then for all but at most N ε primitive characters χ modulo q ≤ N we have (3.1) L (1 , χψ ) = (cid:18) O (cid:18) N (cid:19)(cid:19) Y p ≤ log A N − χ ( p ) ψ ( p ) p ! − . for all primitive characters ψ modulo m . Moreover, if we assume GRH, then (3.1) isvalid with A = 10 , for all primitive characters χ modulo q ≤ N and ψ modulo m . In order to prove this proposition, we first need some preliminary results. Lemma 3.4. Let q be large and χ be a non-principal character modulo q . Let ≤ T ≤ q and X ≥ . Let ≤ σ < and suppose that the rectangle { s : σ < Re( s ) ≤ , | Im( s ) | ≤ T + 3 } does not contain any zeros of L ( s, χ ) . Then we have log L (1 , χ ) = − X p ≤ X log (cid:18) − χ ( p ) p (cid:19) + O (cid:18) log XT + log q (1 − σ ) T + log q log T (1 − σ ) X ( σ − / (cid:19) . Proof. Let α = 1 / log X . Then it follows from Perron’s formula that(3.2) 12 πi Z α + iTα − iT log L (1 + s, χ ) X s s ds = X n ≤ X Λ( n ) n log n χ ( n ) + O ∞ X n =1 Λ( n ) n α log n min (cid:18) , T log | X/n | (cid:19)! = X n ≤ X Λ( n ) n log n χ ( n ) + O (cid:18) log XT + 1 X (cid:19) , ARGE ODD ORDER CHARACTER SUMS 11 by a standard estimation of the error term. Moreover, we observe that X n ≤ X Λ( n ) n log n χ ( n ) = − X p ≤ X log (cid:18) − χ ( p ) p (cid:19) + O ∞ X k =2 X p k >X kp k = − X p ≤ X log (cid:18) − χ ( p ) p (cid:19) + O (cid:16) X − (cid:17) . We now move the contour in (3.2) to the line Re( s ) = σ − 1, where σ = (1 + σ ) / s = 0 that leaves a residue of log L (1 , χ ). Furthermore,it follows from Lemma 8.1 of [7] that for σ ≥ σ and | t | ≤ T we havelog L ( σ + it, χ ) ≪ log qσ − σ ≪ log q − σ . Therefore, we deduce that12 πi Z α + iTα − iT log L (1 + s, χ ) X s s ds = log L (1 , χ ) + E , where E = 12 πi (cid:18)Z σ − − iTα − iT + Z σ − iTσ − − iT + Z α + iTσ − iT (cid:19) log L (1 + s, χ ) X s s ds ≪ log q (1 − σ ) T + log q log T (1 − σ ) X ( σ − / . Since σ ≥ / 2, combining the above estimates completes the proof. (cid:3) Lemma 3.5. Let ξ mod q and ψ mod m be primitive characters. Then, there is aunique primitive character χ such that χψ is induced by ξ if m | q , and no suchcharacter exists if m ∤ q .Proof. Suppose that χψ is induced by ξ , where χ is a primitive character of conductor ℓ . Then we must have q = [ ℓ, m ], and hence there is no such character χ if m ∤ q .Now, suppose that m | q , and let m = p a · · · p a k k be its prime factorization. Weconstruct χ in this case as follows. Since q = [ ℓ, m ], then we have q = q · p b · · · p b k k where ( q , m ) = 1 and b j ≥ a j for all 1 ≤ j ≤ k , and ℓ = q · p c · · · p c k k where c j = b j if b j > a j and 0 ≤ c j ≤ a j if b j = a j . Now, since ξ is primitive then ξ = ˜ ξ · ξ · · · ξ k where ˜ ξ is a primitive character modulo q and ξ j is a primitive character modulo p b j j for 1 ≤ j ≤ k . Similarly, we have ψ = ψ · · · ψ k and χ = ˜ χ · χ · · · χ k where ˜ χ is aprimitive character modulo q and ψ j , χ j are primitive characters modulo p a j j and p c j j respectively. Moreover, since ξ induces χψ then we must have ˜ χ = ˜ ξ , and ξ j induces χ j ψ j for all 1 ≤ j ≤ k . But this implies that χ j ( n ) = ξ j ( n ) ψ j ( n ) for all n such that p j ∤ n , and hence we deduce that there is only one choice for χ j since it is primitive.Since this holds for all 1 ≤ j ≤ k , the character χ is unique. (cid:3) Proof of Proposition 3.3. By Bombieri’s classical zero-density estimate (see Theorem20 of [2]), we know that there are at most N − σ ) (log N ) B primitive characters ξ with conductor q ≤ N log N and such that L ( s, ξ ) has a zero in the rectangle { s : σ ≤ Re( s ) ≤ , | Im( s ) | ≤ N } , where B is an absolute constant. Let ξ , · · · , ξ L be thesecharacters with σ = 1 − ε/ 20. Then, it follows from the above argument that L ≪ N ε/ .Recall that if ξ is a primitive character that induces ˜ ξ , then L ( s, ξ ) and L ( s, ˜ ξ ) havethe same zeros in the half-plane Re( s ) > 0. For a primitive character ψ modulo m , let E ψ denote the set of primitive characters χ modulo q with q ≤ N and such that χψ is induced by one of the characters ξ j for 1 ≤ j ≤ L . Let E m be the union over allprimitive characters ψ modulo m of the sets E ψ . Then, it follows from Lemma 3.5 that |E m | ≤ X ψ mod mψ primitive |E ψ | ≤ Lφ ( m ) ≪ N ε . Let X = (log N ) A where A = 100 /ε . If χ is a primitive character with conductor q ≤ N and such that χ / ∈ E m then it follows from Lemma 3.4 with T = X that for all primitivecharacters ψ modulo m we havelog L (1 , χψ ) = − X p ≤ X log (cid:18) − χ ( p ) ψ ( p ) p (cid:19) + O (cid:18) N (cid:19) , which implies (3.1). Finally, if we assume GRH, then this estimate is valid for allprimitive characters χ modulo q ≤ N and ψ modulo m with X = (log N ) by Lemma3.4. (cid:3) We can now prove Theorem 2.5. Proof of Theorem 2.5. Combining Propositions 3.1 and 3.3 we deduce that for all butat most N ε primitive characters χ modulo q with N ε/ ≤ q ≤ N we have(3.3) M ( χ ) + √ q ≫ √ qmφ ( m ) Y p ≤ log A N − χ ( p ) ψ ( p ) p ! − with A = 100 /ε . The first part of the theorem follows, upon noting that Y p ≤ log A N − χ ( p ) ψ ( p ) p ! − ≫ ε (log q ) · exp (cid:0) − D ( χ, ψ ; log q ) (cid:1) . The second part follows along the same lines, since if we assume GRH then (3.3) holdswith A = 10 for all primitive characters χ with conductor q ≤ N . (cid:3) ARGE ODD ORDER CHARACTER SUMS 13 Estimates for the distance D ( χ, ψ ; y ) : Proofs of Proposition 2.6 andTheorem 1.3 We shall first prove a lower bound for D ( χ, ψ ; y ) , which is a refined version of (2.4),that shows that Proposition 2.6 is best possible. This will also be the main ingredientin the proof of Proposition 2.1. Proposition 4.1. Let g ≥ be a fixed odd integer, and ε > be small. Let ψ be anodd primitive character of conductor m and order k , and y be such that m ≤ (log y ) / .Put k ∗ = k/ ( g, k ) . Then, for any primitive character χ (mod q ) of order g we have D ( χ, ψ ; y ) ≥ (cid:18) − (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) (cid:19) log y − βε log m + O (log m ) , where β = 0 if m is a non-exceptional modulus, and β = 1 if m is exceptional. We say that m ≥ exceptional modulus if there exists a Dirichlet character χ m and a complex number s such that L ( s, χ m ) = 0 and(4.1) Re( s ) ≥ − c log( m (Im( s ) + 2))for some sufficiently small constant c > 0. One expects that there are no such moduli,but what is known unconditionally is that if m is exceptional, then there is only one exceptional character χ m modulo m , which is quadratic, and for which L ( s, χ m ) hasa unique zero in the region (4.1) which is real and simple (this zero is called a Siegelzero).For g ≥ 3, we let µ g denote the set of g -th roots of unity. Then, we observe that D ( χ, ψ ; y ) = log log y − X p ≤ y Re( χ ( p ) ψ ( p )) p + O (1) ≥ log log y − X ℓ mod k max z ∈ µ g ∪{ } Re (cid:18) z · e (cid:18) − ℓk (cid:19)(cid:19) X p ≤ yψ ( p )= e ( ℓk ) 1 p + O (1) . (4.2)Proposition 4.1 follows from this inequality together with Proposition 4.2 below, whichprovides an asymptotic formula for the sum on the right hand side of (4.2). To estab-lish Proposition 2.6, we need an additional ingredient, namely that there exist manyprimitive characters χ whose values we can control at the small primes p ≤ c log q sothat the inequality in (4.2) is sharp for y ≤ c log q . This is proven in Lemma 4.7 below. Proposition 4.2. Let g ≥ be a fixed odd integer, and ε > be small. Let ψ be anodd primitive character of conductor m and order k , and y be such that m ≤ (log y ) / . Put k ∗ = k/ ( g, k ) . Then (4.3) X ℓ mod k max z ∈ µ g ∪{ } Re (cid:18) z · e (cid:18) − ℓk (cid:19)(cid:19) X p ≤ yψ ( p )= e ( ℓk ) 1 p = (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) log y + θε log m + O (log m ) , where θ = 0 if m in a non-exceptional modulus, and | θ | ≤ if m is exceptional. We first record the following lemma, which is a special case of Lemma 8.3 of [4]. Lemma 4.3. Let g, k and k ∗ be as in Proposition 4.2. Then k X ℓ mod k max z ∈ µ g ∪{ } Re (cid:18) z · e (cid:18) − ℓk (cid:19)(cid:19) = (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) . Proof. This is Lemma 8.4 of [4] (see also Lemma 5.2 below) with θ = 0. (cid:3) In view of this lemma, our next task is to estimate the inner sum in the left handside of (4.3). Since ψ is periodic modulo m we have(4.4) X p ≤ yψ ( p )= e ( ℓk ) 1 p = X a mod mψ ( a )= e ( ℓk ) X p ≤ yp ≡ a mod m p . In what follows we shall need estimates of Mertens type for sums of reciprocals ofprimes from specific arithmetic progressions a modulo m that are uniform in a rangeof the modulus m . Results of this type were established by Languasco and Zaccagnini[11]. Lemma 4.4 (Theorem 2 and Corollary 3 of [11]) . Let x ≥ . Then, uniformly in m ≤ log x and reduced residue classes a modulo m , we have − X p ≤ xp ≡ a mod m log (cid:18) − p (cid:19) = 1 φ ( m ) log x − C m ( a ) + O (cid:18) (log log x ) / (log x ) / (cid:19) , where C m ( a ) is defined in (4.5) below. We shall refer to C m ( a ) as the Mertens constant of the residue class a modulo m .Given m ≥ a, m ) = 1, this quantity is defined by(4.5) C m ( a ) := 1 φ ( m ) X χ = χ mod m χ ( a ) · log K (1 , χ ) L (1 , χ ) − φ ( m ) ( γ + log( φ ( m ) /m )) , where, for each non-principal character χ modulo q , K ( s, χ ) := ∞ X n =1 k χ ( n ) n s ARGE ODD ORDER CHARACTER SUMS 15 is an absolutely convergent Dirichlet series for Re( s ) > 0, and k χ ( n ) is a completelymultiplicative function defined as(4.6) k χ ( p ) := p − (cid:18) − χ ( p ) p (cid:19) (cid:18) − p (cid:19) − χ ( p ) ! . Moreover, it follows from the definition of k χ ( p ) and Taylor expansion that(4.7) | k χ ( p ) | ≪ p . In order to study the asymptotic behaviour of the sum in (4.4), it will be crucial tohave an upper bound for the average of | C m ( a ) | . Lemma 4.5. Fix ε > , and let m ≥ . Then, we have X a mod m ( a,m )=1 | C m ( a ) | ≤ ( O (log m ) , if m is a non-exceptional modulus ,ε log m + O (log m ) , if m is exceptional . Proof. First, since φ ( m ) ≫ m/ log m then C m ( a ) = 1 φ ( m ) X χ = χ mod m χ ( a ) · log K (1 , χ ) L (1 , χ ) + O (cid:18) log mφ ( m ) (cid:19) . Let χ be a non-principal character modulo m . By (4.7) we havelog K (1 , χ ) = − X p ≤ x log (cid:18) − k χ ( p ) p (cid:19) + O X p>x | k χ ( p ) | p ! = − X p ≤ x log (cid:18) − k χ ( p ) p (cid:19) + O (cid:18) x (cid:19) . Furthermore, it follows from (4.6) that − log (cid:18) − k χ ( p ) p (cid:19) + log (cid:18) − χ ( p ) p (cid:19) = χ ( p ) log (cid:18) − p (cid:19) . If χ is a non-exceptional character, then L ( σ + it, χ ) does not vanish when σ ≥ − c log( m ( | t | + 2)) , for some positive constant c . Therefore, taking T = m , σ = 1 − c/ (4 log m ) and X = exp((log m ) ) in Lemma 3.4 we obtain(4.8) log L (1 , χ ) = − X p ≤ X log (cid:18) − χ ( p ) p (cid:19) + O (cid:18) m (cid:19) . We first consider the case when m is a non-exceptional modulus. Using the aboveestimates together with the orthogonality of characters we conclude that(4.9) C m ( a ) = 1 φ ( m ) X χ = χ mod m χ ( a ) X p ≤ X χ ( p ) log (cid:18) − p (cid:19) + O (cid:18) log mφ ( m ) (cid:19) = X p ≤ Xp ≡ a mod m log (cid:18) − p (cid:19) − φ ( m ) X p ≤ Xp ∤ m log (cid:18) − p (cid:19) + O (cid:18) log mφ ( m ) (cid:19) . Thus, we deduce in this case that X a mod m ( a,m )=1 | C m ( a ) | ≤ − X a mod m ( a,m )=1 X p ≤ Xp ≡ a mod m log (cid:18) − p (cid:19) − X p ≤ X log (cid:18) − p (cid:19) + O (log m ) ≪ log m. Now, suppose that m is an exceptional modulus, and let χ m be the exceptional charactermodulo m . The approximation (4.8) is valid for all non-principal characters χ = χ m modulo m . Furthermore, for χ = χ m we have Siegel’s bound (see Theorem 11.4 in [14])log L (1 , χ m ) ≥ − ε log m + O ε (1) , and hence, instead of (4.8) we use that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log L (1 , χ m ) + X p ≤ X log (cid:18) − χ m ( p ) p (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε log m + O (log m ) . Thus, similarly to (4.9) we obtain in this case that | C m ( a ) | ≤ − X p ≤ Xp ≡ a mod m log (cid:18) − p (cid:19) + ε log mφ ( m ) + O (cid:18) log mφ ( m ) (cid:19) . Summing over all reduced residue classes a modulo m gives the desired bound. (cid:3) Proposition 4.2 now follows readily. Proof of Proposition 4.2. First, note that for each fixed ℓ modulo k , there are exactly φ ( m ) /k residue classes a modulo m such that ( a, m ) = 1 and ψ ( a ) = e (cid:0) ℓk (cid:1) . This followsfrom the simple fact that the number of such residue classes equals the size of the kernelof ψ , and by basic group theory this is | ( Z /m Z ) ∗ | / | Im( ψ ) | = φ ( m ) /k . Thus, we deducefrom (4.4) and Lemma 4.4 that X p ≤ yψ ( p )= e ( ℓk ) 1 p = X a mod mψ ( a )= e ( ℓk ) log yφ ( m ) − C m ( a ) + X p ≤ yp ≡ a ( m ) (cid:18) log (cid:18) − p (cid:19) + 1 p (cid:19) + O (cid:18) y ) / (cid:19) = log yk − X a mod mψ ( a )= e ( ℓk ) C m ( a ) + X p ≤ yψ ( p )= e ( ℓk ) (cid:18) log (cid:18) − p (cid:19) + 1 p (cid:19) + O (cid:18) φ ( m ) k (log y ) / (cid:19) . ARGE ODD ORDER CHARACTER SUMS 17 Summing over ℓ modulo k , and using Lemma 4.3, we get X ℓ mod k max z ∈ µ g ∪{ } Re (cid:18) z · e (cid:18) − ℓk (cid:19)(cid:19) X p ≤ yψ ( p )= e ( ℓk ) 1 p = (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) log y + θ X a mod m ( a,m )=1 | C m ( a ) | + O (1) , for some complex number | θ | ≤ 1. Appealing to Lemma 4.5 completes the proof. (cid:3) Let ψ be any odd character modulo m , with even order k . In choosing characters χ of order g and conductor q ≤ N that maximize the distance D ( χ, ψ ; y ) with y ≤ (log N ) / 10, we will need to be able to choose the values of χ at the “small” primes p ≤ y .Using Eisenstein’s reciprocity law and the Chinese Remainder Theorem, Goldmakher[4] proved the existence of such characters. Lemma 4.6 (Proposition 9.3 of [4]) . Let g ≥ be fixed, and y be large. Let { z p } p be asequence of complex numbers such that z p ∈ µ g ∪ { } for each prime p . There exists apositive integer q such that g Y p ≤ yp ∤ g p ≤ q ≤ g Y p ≤ yp ∤ g p, and a primitive Dirichlet character χ of order g and conductor q such that χ ( p ) = z p for all p ≤ y with p ∤ g . However, in order to prove Theorem 1.3 we need to find “many” such characters,since we must avoid those in the exceptional set of Theorem 2.5, which has size at most N ε . To this end we prove Lemma 4.7. Let N be large. Let g ≥ be fixed. Let ≤ y ≤ (log N ) / , and put z = ( z p ) p ≤ y ∈ ( µ g ∪ { } ) π ( y ) . There are ≫ N / g π ( y )+2 log N primitive Dirichlet characters χ of order g and conductor q ≤ N such that χ ( p ) = z p for each p ≤ y such that p ∤ g . The special case z = = (1 , , . . . , 1) was proved by the first author in Lemma2.3 of [10], but the proof there does not appear to generalize to all z ∈ ( µ g ∪ { } ) π ( y ) .However, we will show that one can combine the special case z = with Lemma 4.6in order to obtain the general case in Lemma 4.7. Proof of Lemma 4.7. Let S z ,g ( N ) be the set of all characters χ of order g and conductor q ≤ N such that χ ( p ) = z p for all p ≤ y with p ∤ g . By Lemma 4.6, there exists ℓ and a primitive Dirichlet character ξ of order g and conductor ℓ such that ξ ( p ) = z p for all p ≤ y with p ∤ g . Moreover, one haslog ℓ = X p ≤ y log p + O g (1) = y (1 + o (1)) , by the prime number theorem, and hence ℓ ≤ N / by our assumption on y .On the other hand, Lemma 2.3 of [10] implies that there are ≫ N / g π ( y )+2 log N primitive Dirichlet characters ψ n of order g and conductor n , such that n = q q where N / < q < q < N / are primes with p ≡ p ≡ g , and such that ψ n ( p ) = 1for all primes p ≤ y . Now, for any such n we have ( ℓ, n ) = 1 since ℓ ≤ N / , and hence ψ n ξ is a primitive character of order g and conductor nℓ ≤ N . Finally observe that ψ n ξ ( p ) = z p for each p ≤ y such that p ∤ g . Thus we deduce that ψ n ξ ∈ S z ,g ( N ) forevery character ψ n , completing the proof. (cid:3) We finish this section by proving Proposition 2.6 and Theorem 1.3. Proof of Proposition 2.6. Let m be a non-exceptional modulus, and ψ be an odd prim-itive character modulo m with order k . For each 0 ≤ ℓ ≤ k − 1, suppose that themaximum of Re (cid:0) ze (cid:0) − ℓk (cid:1)(cid:1) for z ∈ ( µ g ∪ { } ) π ( y ) is attained when z = z ℓ . Then, itfollows from Lemma 4.7 that there are at least √ N primitive characters χ of order g and conductor q ≤ N such that X p ≤ y Re χ ( p ) ψ ( p ) p = X ℓ mod k Re (cid:18) z ℓ · e (cid:18) − ℓk (cid:19)(cid:19) X p ≤ yψ ( p )= e ( ℓk ) 1 p + O g (1) . The desired result then follows from (4.2) and Proposition 4.2. (cid:3) Proof of Theorem 1.3. Let N be sufficiently large, and let y = (log N ) / 10. Let m bea prime number that is also a non-exceptional modulus, such that p log N ≤ m ≤ p log N . One can make such a choice since it is known that there is at most oneexceptional prime modulus between x and 2 x for any x ≥ ψ be a primitive character modulo m of order k = φ ( m ) = m − 1. Notethat such a character is necessarily odd. By Proposition 2.6, there are at least √ N / g and conductor N / ≤ q ≤ N such that D ( χ, ψ ; y ) = (cid:18) − (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) (cid:19) log y + O (log m ) = δ g log q + O (log q ) , since gk ∗ ≥ k and t/ tan( t ) = 1+ O ( t ). Thus, since D ( χ, ψ ; log q ) = D ( χ, ψ ; y ) + O (1),then it follows from Theorem 2.5 (with ε = 1 / 4) that there are at least √ N / ARGE ODD ORDER CHARACTER SUMS 19 characters of order g and conductor N / ≤ q ≤ N such that M ( χ ) ≫ √ qmφ ( m ) (log q ) − δ g (log q ) O (1) ≫ √ q (log q ) − δ g (log q ) − (log q ) O (1) . (cid:3) Estimates for M ( χψ ; y, T ) : Proofs of Propositions 2.1 and 2.2 Lower bounds for M ( χψ ; y, T ) for small twists T : Proof of Proposition2.1. Let χ be a primitive character modulo q of odd order g ≥ 3, and ψ be an oddprimitive character of conductor m and order k . Let y ≥ exp( m / (4 α ) ) be a real number,and put z = exp ((log y ) α ). Since m ≤ (log z ) / , then it follows from Proposition 4.1that for all x ≥ z we have(5.1) D ( χ, ψ ; x ) ≥ (cid:18) − (1 − δ g ) π/gk ∗ tan( π/gk ∗ ) (cid:19) log x − βε log m + O (log m ) ≥ (cid:18) δ g + π (1 − δ g )4( gk ∗ ) (cid:19) log x − βε log m + O (log m ) . since gk ∗ ≥ 6, and u/ tan( u ) ≤ − u / ≤ u ≤ π/ t be a real number such that | t | ≤ (log y ) − α . First, if | t | ≤ (log y ) − , then since p − it = 1 + O ( | t | log p ) we obtain(5.2) D ( χψ, n it ; y ) = D ( χ, ψ ; y ) + O | t | X p ≤ y log pp ! = D ( χ, ψ ; y ) + O (1) , and hence the desired lower bound for D ( χψ, n it ; y ) follows from (5.1). Thus, we canand will assume throughout this subsection that | t | > (log y ) − . Furthermore, since | t | ≤ (log y ) − α = 1 / log z , then similarly to (5.2) one has D ( χψ, n it ; y ) = D ( χ, ψ ; z ) + X z
Proposition 5.1. Let χ , ψ , y , z and t be as above. Then we have X z
Lemma 5.2 (Lemma 8.3 of [4]) . Let g ≥ be odd, k ≥ be even, and θ ∈ R . Put k ∗ = k/ ( g, k ) . Then we have k X ℓ mod k max z ∈ µ g ∪{ } Re (cid:18) z · e (cid:18) θ − ℓk (cid:19)(cid:19) = sin( π/g ) k ∗ tan( π/gk ∗ ) F gk ∗ ( − gk ∗ θ ) , where F n ( u ) := cos(2 π { u } /n ) + tan( π/n ) sin(2 π { u } /n ) , and { u } is the fractional partof u . Lemma 5.3. Let T > and n ≥ be a positive integer. Then (5.3) Z T F n ( u ) u du = nπ tan (cid:16) πn (cid:17) log T + O (1) , and (5.4) Z /T F n ( u ) u du = log T + O (1) . In particular, for any < A < B we have (5.5) Z BA F n ( u ) u du ≤ nπ tan (cid:16) πn (cid:17) log( B/A ) + O (1) , and the constants in the O (1) error terms are absolute.Proof. We first prove (5.3). Since F n is bounded and 1-periodic, we have Z T F n ( u ) u du = X ≤ j ≤ T Z F n ( u ) u + j du + O (1) = X ≤ j ≤ T j Z F n ( u ) du + O (1)= nπ tan (cid:16) πn (cid:17) log T + O (1) . The second estimate (5.4) follows from observing that for u ∈ [0 , 1) and n ≥ F n ( u ) = 1 + O (cid:18) u n + tan (cid:16) πn (cid:17) un (cid:19) = 1 + O ( u ) . Finally, to prove (5.5) we consider the three cases 1 ≤ A < B , A < < B , and A < B ≤ 1. The first case follows from (5.3), and the third follows from (5.4) uponusing the inequality tan( π/n ) ≥ π/n . Finally, in the second case we have Z BA F n ( u ) u du = Z A F n ( u ) u du + Z B F n ( u ) u du = nπ tan (cid:16) πn (cid:17) log B − log A + O (1) , which implies the result since tan( π/n ) ≥ π/n and − log A > . (cid:3) Proof of Proposition 5.1. Let x = z , and δ > r ≤ R := ⌊ log( y/z ) / log(1 + δ ) ⌋ , set x r := (1 + δ ) r z . We considerthe sum S = X z
For each 0 ≤ r ≤ R − 1, we define S r := X x r
1. Moreover, for x r < p ≤ x r +1 = (1 + δ ) x r , we have1 p = log px r log x r (cid:18) p log p − x r log x r x r log x r (cid:19) − = (cid:0) O ( δ ) (cid:1) log px r log x r . Thus, combining these two statements, we get X xr
1, where the constantin the O is absolute. This shows that for all u ∈ [log x r , log x r +1 ) we have F gk ∗ (cid:18) tgk ∗ π u (cid:19) = F gk ∗ (cid:18) tgk ∗ π log x r (cid:19) + O (cid:0) δ | t | (cid:1) . Furthermore, we note that Z log x r +1 log x r duu = (cid:0) O ( δ ) (cid:1) δ log x r . Combining these two facts, we obtain δ log x r F gk ∗ (cid:18) tgk ∗ π log x r (cid:19) = (cid:0) O ( δ ) (cid:1) Z log x r +1 log x r F gk ∗ (cid:18) tgk ∗ π log x r (cid:19) duu = (cid:0) O ( δ ) (cid:1) Z log x r +1 log x r F gk ∗ (cid:18) tgk ∗ π u (cid:19) duu + O (cid:18) δ | t | Z log x r +1 log x r duu (cid:19) . Summing over 0 ≤ r ≤ R − 1, we get δ X ≤ r ≤ R − F gk ∗ (cid:0) tgk ∗ π log x r (cid:1) log x r = (cid:0) O ( δ ) (cid:1) Z log y log z F gk ∗ (cid:18) tgk ∗ π u (cid:19) duu + O ( δ ) , since R log y log x R dt/t ≪ δ .We now estimate the integral in the main term above. One can easily check that for n ≥ F n is an even function. Making the change of variable v := gk ∗ | t | π u , and setting A := gk ∗ | t | π log z and B := gk ∗ | t | π log y , we get Z log y log z F gk ∗ (cid:18) tgk ∗ π u (cid:19) duu = Z BA F gk ∗ ( v ) dvv ≤ gk ∗ π tan (cid:18) πgk ∗ (cid:19) log (cid:18) log y log z (cid:19) + O (1) , by Lemma 5.3. Combining the above estimates with (5.6) and (5.7) we obtain S ≤ (cid:0) O ( δ ) (cid:1) gπ sin (cid:18) πg (cid:19) log (cid:18) log y log z (cid:19) + O (1) ≤ (1 − δ g ) log (cid:18) log y log z (cid:19) + O ( δ log y ) . Choosing δ = (log y ) − completes the proof of Proposition 5.1. Proposition 2.1 followsupon combining this result with (5.1). (cid:3) Estimating M ( χψ ; y, T ) for large twists T . In this subsection, we prove thefollowing result which implies Proposition 2.2. Proposition 5.4. Assume GRH. Let g ≥ be a fixed odd integer. Let N be largeand y ≤ (log N ) / . Let ψ be an odd primitive character of conductor m such that exp (cid:0) p log y (cid:1) ≤ m ≤ exp (cid:0) √ log y (cid:1) . Then, there exist at least √ N primitive characters χ of order g and conductor q ≤ N such that D ( χψ, n i ; y ) = δ g log y + O (log m ) . Proof. We follow the proof of Proposition 5.1 in such a way that we achieve equalityin all steps. Since the arguments here are similar to those in that proof, we omit someof the details.Let z := exp ((log m ) ) and y ≥ z . Let δ > R := ⌊ log( y/z ) / log(1 + δ ) ⌋ as before. Set x = z and x r := (1 + δ ) r x . Then, as P p ≤ z p ≪ log m , it suffices to find at least √ N primitive characters χ of order g andconductor q ≤ N such that(5.8) X z
Let θ r := − log x r π , for each 0 ≤ r ≤ R − 1. As in the proof of Proposition 5.1, when x r < p ≤ x r +1 we approximate p i by x ir , for each 0 ≤ r ≤ R − 1. Let k be the order of ψ , and for each r let { z r,ℓ } ℓ ∈ ( µ g ∪ { } ) k be chosen so as to maximize the sum X ℓ mod k Re (cid:18) z r,ℓ · e (cid:18) θ r − ℓk (cid:19)(cid:19) X a mod mψ ( a )= e ( ℓ/k ) X xr
The key ingredient to the proof of Theorem 1.4 is the following generalization ofTheorem 2 of [13]. Theorem 6.1. Let f ∈ F and x ≥ . Then, for any < T ≤ we have X n ≤ x f ( n ) n ≪ x Z / log x H T ( α ) α dα, where H T ( α ) = ∞ X k = −∞ max s ∈A k,T ( α ) (cid:12)(cid:12)(cid:12)(cid:12) F (1 + s ) s (cid:12)(cid:12)(cid:12)(cid:12) ! / . and A k,T ( α ) = { s = σ + it : α ≤ σ ≤ , | t − kT | ≤ T / } . Montgomery and Vaughan [13] established this result for T = 1, and a straightfor-ward generalization of their proof allows one to obtain Theorem 6.1 for any 0 < T ≤ Lemma 6.2. Let < α, T ≤ . Then we have (6.1) Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + α + it ) α + it (cid:12)(cid:12)(cid:12)(cid:12) dt + Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F (1 + α + it )( α + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt ≪ H T ( α ) α . ARGE ODD ORDER CHARACTER SUMS 25 Proof. First, we have Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + α + it ) α + it (cid:12)(cid:12)(cid:12)(cid:12) dt = ∞ X k = −∞ Z kT + T/ kT − T/ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + α + it ) α + it (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ ∞ X k = −∞ max | t − kT |≤ T/ (cid:12)(cid:12)(cid:12)(cid:12) F (1 + α + it ) α + it (cid:12)(cid:12)(cid:12)(cid:12) Z kT + T/ kT − T/ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + α + it ) F (1 + α + it (cid:12)(cid:12)(cid:12)(cid:12) dt. To bound the integral on the right hand side of this inequality, we appeal to a result ofMontgomery (see Lemma 6.1 of [17]) which states that if P n ≥ a n n − s and P n ≥ b n n − s are two Dirichlet series which are absolutely convergent for Re( s ) > | a n | ≤ b n for all n ≥ 1, then we have(6.2) Z u − u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 a n n σ + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z u − u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 b n n σ + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt, for any real numbers u ≥ σ > 1. This implies that Z kT + T/ kT − T/ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + α + it ) F (1 + α + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt = Z T/ − T/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 Λ( n ) f ( n ) n α + ikT + it (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≪ Z T/ − T/ (cid:12)(cid:12)(cid:12)(cid:12) ζ ′ (1 + α + it ) ζ (1 + α + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt ≪ Z T/ − T/ | α + it | dt ≤ Z ∞−∞ α + t dt ≪ α . Hence, we deduce that Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + α + it ) α + it (cid:12)(cid:12)(cid:12)(cid:12) dt ≪ H T ( α ) α . To complete the proof, note that Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F (1 + α + it )( α + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ ∞ X k = −∞ max | t − kT |≤ T/ (cid:12)(cid:12)(cid:12)(cid:12) F (1 + α + it ) α + it (cid:12)(cid:12)(cid:12)(cid:12) Z kT + T/ kT − T/ | α + it | dt ≪ H T ( α ) α . (cid:3) Proof of Theorem 6.1. Let S ( x ) = X n ≤ x f ( n ) n . From the Euler product, | F (2) | > 0, so H T ( α ) ≫ 1. Thus, it is enough to prove thestatement for x ≥ x , where x is a suitably large constant. Moreover, observe that R / log x H T ( α ) α − dα is strictly increasing as a function of x , and | S ( x ) log x | is strictlyincreasing for x ∈ [ n, n + 1), for all n ≥ 1. Hence it is enough to prove the result for x ∈ B where B = { x ≥ x : | S ( y ) log y | < | S ( x ) log x | for all y < x } . Montgomery and Vaughan proved that for x ∈ B we have (see equations (7) and (8)of [13]) | S ( x ) | log x ≪ Z xe | S ( u ) | u du + 1log x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x f ( n ) n (log n ) log (cid:16) xn (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1log x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x f ( n ) n log (cid:16) xn (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Integrating the first integral by parts, we get(6.3) Z xe | S ( u ) | u du ≪ J ( x )log x + Z xe J ( u ) u (log u ) du, where J ( u ) := Z ue | S ( t ) | log tt dt ≪ (log u ) / (cid:18)Z ue | S ( t ) | (log t ) t dt (cid:19) / , by the Cauchy-Schwarz inequality. Using Parseval’s Theorem, Montgomery and Vaughanproved that (see equation (14) of [13]) Z ue | S ( t ) | (log t ) t dt ≪ Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + β + it ) β + it (cid:12)(cid:12)(cid:12)(cid:12) dt + Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F (1 + β + it )( β + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt, where β = 2 / log u . Appealing to Lemma 6.2 and making the change of variable α =1 / log u in the integral of the right hand side of (6.3) we deduce that(6.4) Z xe | S ( u ) | u du ≪ H T (cid:18) x (cid:19) + Z / log x H T (2 α ) α dα. Since H T ( α ) is decreasing as a function of α , we have(6.5) H T (cid:18) x (cid:19) ≪ Z / log x / log x H T ( α ) α dα ≤ Z / log x H T ( α ) α dα. Combining (6.4) and (6.5) we get Z xe | S ( u ) | u du ≪ Z / log x H T ( α ) α dα. Furthermore, Montgomery and Vaughan proved that (see pages 207-208 of [13]) X n ≤ x f ( n ) n (log n ) log (cid:16) xn (cid:17) ≪ β Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F ′ (1 + β + it ) β + it (cid:12)(cid:12)(cid:12)(cid:12) dt ! / and X n ≤ x f ( n ) n log (cid:16) xn (cid:17) ≪ β Z ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12) F (1 + β + it )( β + it ) (cid:12)(cid:12)(cid:12)(cid:12) dt ! / , where β = 2 / log x . Combining these bounds with Lemma 6.2 and equation (6.5)completes the proof. (cid:3) ARGE ODD ORDER CHARACTER SUMS 27 In order to derive Theorem 1.4 from Theorem 6.1, we need to bound H T ( α ), andhence to bound | F (1 + s ) | for Re( s ) ≥ α . Tenenbaum (see Section III.4 of [17]) provedthat for all y, T ≥ 2, and 1 / log y ≤ α ≤ 1, we have(6.6) max | t |≤ T | F (1 + α + it ) | ≪ (log y ) exp (cid:0) − M ( f ; y, T ) (cid:1) . However, this bound does not hold for all T > / log y ≤ α ≤ 1. Indeed,taking f to be the M¨obius function µ , α = 1 / y large and T = 1 / log y shows thatmax | t |≤ T | F (1 + α + it ) | ≥ | ζ (3 / | − , while M ( f ; y, T ) = min | t |≤ / log y X p ≤ y p − it ) p = 2 X p ≤ y p + O (1) = 2 log log y + O (1) , and hence the right side of (6.6) is ≪ / (log y ). Nevertheless, using Tenenbaum’s ideas,we show that (6.6) is valid whenever T ≥ α . Lemma 6.3. Let y ≥ and f ∈ F such that f ( p ) = 0 for p > y . Let F ( s ) be itscorresponding Dirichlet series. Then, for all real numbers < α ≤ and T ≥ α wehave max | t |≤ T (cid:12)(cid:12) F (1 + α + it ) | ≪ (log y ) exp (cid:0) − M ( f ; y, T ) (cid:1) . Proof. Note that(6.7) M ( f ; y, T ) = log y − max | t |≤ T Re X p ≤ y f ( p ) p it + O (1) . We first remark that the result is trivial if α ≤ / log y , since in this case we havelog | F (1 + α + it ) | = Re X p ≤ y f ( p ) p α + it + O (1) = Re X p ≤ y f ( p ) p it + O (1) , which follows from the fact that | p α − | ≪ α log p .Now, suppose that α ≥ / log y and put A = exp(1 /α ). Then we havelog | F (1+ α + it ) | = Re X p ≤ y f ( p ) p α + it + O (1) = Re X p ≤ A f ( p ) p α + it + O (1) = Re X p ≤ A f ( p ) p it + O (1) , since P p>A p − − α ≪ | β | ≤ α/ X p ≤ A f ( p ) p i ( t + β ) = X p ≤ A f ( p ) p it + O (1) , and hence max | t |≤ T (cid:12)(cid:12) F (1 + α + it ) | ≪ max | t |≤ ( T − α/ exp Re X p ≤ A f ( p ) p it ! . Now, let | t | ≤ T − α/ Z t + α/ t − α/ Re X p ≤ y f ( p ) p iu ! du = Re X p ≤ y f ( p ) p it (cid:18) p iα/ − p − iα/ i log p (cid:19) = α Re X p ≤ A f ( p ) p it ! + O α + X p>A p log p ! . Since P p>A ( p log p ) − ≪ α by the prime number theorem, we deduce thatRe X p ≤ A f ( p ) p it = 1 α Z t + α/ t − α/ Re X p ≤ y f ( p ) p iu ! du + O (1) ≤ max | t |≤ T Re X p ≤ y f ( p ) p it + O (1) . Appealing to (6.7) completes the proof. (cid:3) We finish this section by proving a slightly stronger form of Theorem 1.4, which weshall need to prove Theorems 1.1 and 1.2. One can also show that the following resultfollows from Theorem 1.4, so it is in fact equivalent to it. Theorem 6.4. Let f ∈ F and x, y ≥ be real numbers. Then, for any real number < T ≤ we have X n ≤ xn ∈S ( y ) f ( n ) n ≪ (log y ) · exp (cid:0) − M ( f ; y, T ) (cid:1) + 1 T , where the implicit constant is absolute, and S ( y ) is the set of y -friable numbers.Proof. First, observe that the result is trivial if T ≤ / log x , since we have in this case X n ≤ xn ∈S ( y ) f ( n ) n ≪ X n ≤ x n ≪ log x ≪ T . Now assume that 1 / log x < T ≤ 1. Let g be the completely multiplicative functionsuch that g ( p ) = f ( p ) for p ≤ y and g ( p ) = 0 otherwise, and let G be its correspondingDirichlet series. Then, it follows from Theorem 6.1 that(6.8) X n ≤ xn ∈S ( y ) f ( n ) n = X n ≤ x g ( n ) n ≪ x Z / log x H T ( α ) α dα, where H T ( α ) = ∞ X k = −∞ max s ∈A k,T ( α ) (cid:12)(cid:12)(cid:12)(cid:12) G (1 + s ) s (cid:12)(cid:12)(cid:12)(cid:12) ! / . First, observe that if | t − kT | ≤ T / k = 0 then | t | ≍ | k | T . Moreover, uniformlyfor all t ∈ R , we have(6.9) | G (1 + σ + it ) | ≤ ζ (1 + σ ) ≪ σ . ARGE ODD ORDER CHARACTER SUMS 29 We will first bound H T ( α ) when α > T . Using (6.9) we obtain in this case(6.10) α · H T ( α ) ≪ ∞ X k = −∞ max | t − kT |≤ T/ α + t ≪ X | k | >α/T k T + X | k |≤ α/T α ≪ αT . Now, suppose that 0 < α ≤ T . To bound H T ( α ) in this case, we first use (6.9) for | k | ≥ 1. This gives H T ( α ) ≪ α X | k |≥ k T + 1 α max s ∈A ,T ( α ) | G (1 + s ) | ≪ αT ) + 1 α max s ∈A ,T ( α ) | G (1 + s ) | . Furthermore, by (6.9) and Lemma 6.3 we havemax s ∈A ,T ( α ) | G (1 + s ) | ≪ max | t |≤ Tσ ≥ T | G (1 + σ + it ) | + max | t |≤ Tα ≤ σ ≤ T | G (1 + σ + it ) |≪ T + (log y ) exp (cid:0) − M ( g ; y, T ) (cid:1) . Since M ( g ; y, T ) = M ( f ; y, T ) we deduce that for 0 < α ≤ T we have(6.11) H T ( α ) ≪ αT ) + (log y ) α exp ( − M ( f ; y, T )) . Using (6.10) when T < α ≤ / log x ≤ α ≤ T we get Z / log x H T ( α ) α dα ≪ (cid:18) T + log y · exp ( −M ( f ; y, T )) (cid:19) Z T / log x α dα + 1 T / Z T α / dα ≪ log xT + (log x )(log y ) exp ( −M ( f ; y, T )) . Inserting this bound in (6.8) yields the result. (cid:3) Proofs of Theorems 2.3, 1.1 and 1.2 To prove Theorem 2.3, the general strategy we use is that of [9] (with the refinementsfrom [4]), and it will be clear where we shall make use of Theorem 6.4. We will considerthe conditional (on GRH) and unconditional results simultaneously, setting y := log q if we are assuming GRH, and setting y := q otherwise. We recall here that y = Q in theunconditional case, and y = Q on GRH, so that in all cases we have log y ≍ log Q .When χ is primitive and α ∈ R , we have X n ≤ q χ ( n ) n e ( nα ) = X n ≤ qn ∈S ( y ) χ ( n ) n e ( nα ) + O (1); on GRH, this follows from (2.5), and unconditionally this statement is trivial. Insertingthis estimate in P´olya’s Fourier expansion (2.1) gives M ( χ ) ≪ √ q max α ∈ [0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤| n |≤ qn ∈S ( y ) χ ( n ) n (cid:0) − e ( nα ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1 . Therefore, to prove Theorem 2.3 it suffices to show that for all α ∈ [0 , 1] we have(7.1) X ≤| n |≤ qn ∈S ( y ) χ ( n ) n e ( nα ) ≪ (cid:16) − χ ( − ξ ( − (cid:17) √ mφ ( m ) (log Q ) e −M ( χξ ; Q, (log Q ) − / ) +(log Q ) + o (1) . Let α ∈ [0 , 1] and R := (log Q ) . By Dirichlet’s theorem on Diophantine approximation,there exists a rational approximation | α − b/r | ≤ /rR , with 1 ≤ r ≤ R and ( b, r ) = 1.Let M := (log Q ) / . We shall distinguish between two cases. If r ≤ M , we say that α lies on a major arc, and if M < r ≤ R we say that α lies on a minor arc. In thelatter case, we shall use Corollary 2.2 of [4], which is a consequence of the work ofMontgomery and Vaughan [15]. Indeed, this shows that X ≤| n |≤ qn ∈S ( y ) χ ( n ) n e ( nα ) ≪ (log M ) / √ M log y + log R + log y ≪ (log Q ) + o (1) . We now handle the more difficult case of α lying on a major arc. First, it followsfrom Lemma 4.1 of [4] (which is a refinement of Lemma 6.2 of [9]) that for N :=min { q, | rα − b | − } , we have(7.2) X ≤| n |≤ qn ∈S ( y ) χ ( n ) n e ( nα ) = X ≤| n |≤ Nn ∈S ( y ) χ ( n ) n e (cid:18) nbr (cid:19) + O (cid:18) (log R ) / √ R (log y ) + log R + log y (cid:19) = X ≤| n |≤ Nn ∈S ( y ) χ ( n ) n e (cid:18) nbr (cid:19) + O (log Q ) . We first assume that b = 0. In this case we can use an identity of Granville andSoundararajan (see Proposition 2.3 of [4]) which asserts that(7.3) X ≤| n |≤ Nn ∈S ( y ) χ ( n ) n e (cid:18) nbr (cid:19) = (cid:16) − χ ( − ψ ( − (cid:17) X d | rd ∈S ( y ) χ ( d ) d · φ ( r/d ) X ψ mod r/d τ ( ψ ) ψ ( b ) X n ≤ N/dn ∈S ( y ) χ ( n ) ψ ( n ) n . ARGE ODD ORDER CHARACTER SUMS 31 To bound the inner sum above, we appeal to Theorem 6.4 with T = (log Q ) − / . Thisimplies that X n ≤ N/dn ∈S ( y ) χ ( n ) ψ ( n ) n ≪ (log y ) · exp (cid:0) −M ( χψ ; y, (log Q ) − / ) (cid:1) + (log Q ) / . Moreover, in the conditional case y = Q , and thus we have M ( χψ ; y, (log Q ) − / ) ≥ M ( χψ ; Q, (log Q ) − / ) + O (1) . Therefore, we get(7.4) X n ≤ N/dn ∈S ( y ) χ ( n ) ψ ( n ) n ≪ (log Q ) · exp (cid:0) −M ( χψ ; Q, (log Q ) − / ) (cid:1) + (log Q ) / . We now order the primitive characters ψ (mod ℓ ) for ℓ ≤ M (including the trivialcharacter ψ which equals 1 for all integers) as { ψ k } k , where M ( χψ k ; Q, (log Q ) − / ) ≤ M ( χψ k +1 ; Q, (log Q ) − / ) , for all k ≥ 1. Note that ψ = ξ , in the notation of Theorem 2.3. Furthermore, by aslight variation of Lemma 3.1 of [1] we have M (cid:0) χψ k ; Q, (log Q ) − / (cid:1) ≥ (cid:18) − √ k (cid:19) log Q + O (cid:16)p log Q (cid:17) . Therefore, if ψ (mod ℓ ) is induced by ψ k , then(7.5) M (cid:0) χψ ; Q, (log Q ) − / (cid:1) ≥ M (cid:0) χψ k ; Q, (log Q ) − / (cid:1) + O X p | ℓ p ≥ (cid:18) − √ k + o (1) (cid:19) log Q, since P p | ℓ /p ≪ log ℓ ≪ log Q . Inserting this bound in (7.4), we deduce that thecontribution of all characters ψ that are induced by some ψ k with k ≥ ≪ (log Q ) / X d | r dφ ( r/d ) X ψ mod r/d | τ ( ψ ) | ≪ (log Q ) / X d | r √ rd / ≪ (log Q ) / , since 1 / √ < / | τ ( ψ ) | ≤ p r/d , and r ≤ (log Q ) / . Moreover, observe that thereis at most one character ψ (mod r/d ) such that ψ is induced by ψ . Using (7.5), wededuce that the contribution of these characters to (7.3) is ≪ (log Q ) / √ o (1) X d | r d · p r/dφ ( r/d ) ≪ (log Q ) / √ o (1) log r ≪ (log Q ) / √ o (1) . Thus, it now remains to estimate the contribution of the characters ψ mod r/d that areinduced by ξ , recalling that ξ has conductor m . If m ∤ r , there are no such characters ψ and the theorem follows in this case. If m | r and ψ mod r/d is induced by ξ , thenwe must have d | ( r/m ). Furthermore, by Lemma 4.1 of [9] we have τ ( ψ ) = µ (cid:16) rdm (cid:17) ξ (cid:16) rdm (cid:17) τ ( ξ ) . Therefore, the contribution of these characters to (7.3) is(7.6) (cid:16) − χ ( − ξ ( − (cid:17) ξ ( b ) τ ( ξ ) X d | ( r/m ) d ∈S ( y ) χ ( d ) d · φ ( r/d ) µ (cid:16) rdm (cid:17) ξ (cid:16) rdm (cid:17) X n ≤ N/d ( n,r/d )=1 n ∈S ( y ) χ ( n ) ξ ( n ) n . Furthermore, it follows from Lemma 4.4 of [9] that X n ≤ N/d ( n,r/d )=1 n ∈S ( y ) χ ( n ) ξ ( n ) n = X n ≤ N ( n,r/d )=1 n ∈S ( y ) χ ( n ) ξ ( n ) n + O (log d )= Y p | rd (cid:18) − χ ( p ) ξ ( p ) p (cid:19) X n ≤ Nn ∈S ( y ) χ ( n ) ξ ( n ) n + O (cid:0) (log Q ) (cid:1) . Thus, in view of Theorem 6.4, we deduce that (7.6) is(7.7) ≪ (cid:16) − χ ( − ξ ( − (cid:17) √ m (cid:16) (log Q ) e −M ( χξ ; Q, (log Q ) − / ) + (log Q ) / (cid:17) × X d | ( r/m )( r/ ( dm ) ,m )=1 dφ ( r/d ) µ (cid:16) rdm (cid:17) Y p | rdm (cid:18) p (cid:19) . Finally, by a change of variables a = r/ ( md ), we obtain X d | ( r/m )( r/ ( dm ) ,m )=1 dφ ( r/d ) µ (cid:16) rdm (cid:17) Y p | rdm (cid:18) p (cid:19) = mrφ ( m ) X a | ( r/m )( a,m )=1 aφ ( a ) µ ( a ) Y p | a (cid:18) p (cid:19) ≤ φ ( m ) · r/m Y p | ( r/m ) (cid:18) p + 1 p − (cid:19) ≤ φ ( m ) , since 2 p/ ( p − ≤ p for all primes p ≥ 3. Combining this bound with (7.7), it followsthat the contribution of the characters ψ that are induced by ξ to (7.3) is ≪ (cid:16) − χ ( − ξ ( − (cid:17) √ mφ ( m ) (log Q ) e −M ( χξ ; Q, (log Q ) − / ) + (log Q ) / . It thus remains to consider when b = 0, and hence r = 1. First, if ξ is identically 1(so m = 1), then a trivial application of Theorem 6.4 shows that in this case X ≤| n |≤ Nn ∈S ( y ) χ ( n ) n ≪ (cid:16) − χ ( − (cid:17) √ mφ ( m ) (log Q ) e −M ( χ ; Q, (log Q ) − / ) + (log Q ) / . ARGE ODD ORDER CHARACTER SUMS 33 On the other hand, if ξ is not the trivial character, then it follows from (7.5) that M ( χ ; Q, (log Q ) − / ) ≥ (cid:18) − √ o (1) (cid:19) log Q, and hence by Theorem 6.4 we get X ≤| n |≤ Nn ∈S ( y ) χ ( n ) n ≪ (log Q ) / √ o (1) , which completes the proof of (7.1). Theorem 2.3 follows as well.We end this section by deducing Theorems 1.1 and 1.2 from Theorem 2.3 andProposition 2.1. We shall prove both results simultaneously, by setting Q := log q onGRH and Q := q unconditionally. Proof of Theorems 1.1 and 1.2. Let ξ be the character of conductor m ≤ (log Q ) / that minimizes M (cid:0) χψ ; Q, (log Q ) − / (cid:1) . If ξ is even, then it follows from Theorem 2.3that M ( χ ) ≪ √ q (log Q ) / o (1) , which trivially implies the result in this case since 1 − δ g > / 11, for all g ≥ 3. Now,suppose that ξ is odd and let k be its order. We also let β = 1 if m is an exceptionalmodulus, and β = 0 otherwise. Then, combining Theorem 2.3 and Proposition 2.1(with α = 7 / 11) we obtain(7.8) M ( χ ) ≪ √ qmφ ( m ) (log Q ) − δ g exp (cid:18) − c (1 − δ g )( gk ∗ ) log Q + βε log m + O (log m ) (cid:19) ≪ √ q (log Q ) − δ g exp (cid:18) − (cid:18) − βε (cid:19) log m − c (1 − δ g ) g m log Q + c log m (cid:19) , for some positive constants c , c , since φ ( m ) ≫ m/ log m . One can easily check thatthe expression inside the exponential is maximal when m ≍ p log Q , and its maximumequals − (cid:18) − βε (cid:19) log Q + O (log Q ) . Inserting this estimate in (7.8) completes the proof. (cid:3) References [1] A. Balog, A. Granville and K. Soundararajan Multiplicative functions in arithmetic progres-sions. Ann. Math. Qu. 37 (2013), no. 1, 3–30.[2] E. Bombieri, On the Large Sieve. Mathematika, 12 (1965), 201–225.[3] H. Davenport, Multiplicative number theory . Third edition. Revised and with a preface byHugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000.xiv+177 pp.[4] L. Goldmakher, Multiplicative mimicry and improvements of the P´olya-Vinogradov inequal-ity. Algebra Number Theory 6 (2012), no. 1, 123–163.[5] L. Goldmakher and Y. Lamzouri, Lower bounds on odd order character sums , Int. Math.Res. Not. 2012 (2012), no. 21, 5006–5013. [6] L. Goldmakher and Y. Lamzouri, Large even order character sums , Proc. Amer. Math. Soc.142 (2014), no. 8, 2609–2614.[7] A. Granville and K. Soundararajan, Large character sums , J. Amer. Math. Soc. 14 (2001),no. 2, 365–397.[8] A. Granville and K. Soundararajan, The distribution of values of L (1 , χ d ) . Geom. Funct.Anal. 13 (2003), no. 5, 992–1028.[9] A. Granville and K. Soundararajan, Large character sums: pretentiuous characters and theP´olya-Vinogradov theorem Jour. AMS 20 (2007), no. 2, 357-384.[10] Y. Lamzouri, Large Values of L (1 , χ ) for k th order characters χ and applications to charactersums , Mathematika 63 (2017), no. 1, 53-71.[11] A. Languasco and A. Zaccagnini, A note on Mertens’ formula for arithmetic progressions. J. Number Theory, 127 (2007), no. 1, 37–46.[12] H. L. Montgomery, Topics in multiplicative number theory , Lecture Notes in Mathematics,Vol. 227. Springer-Verlag, Berlin-New York, 1971.[13] H. L. Montgomery and R.C. Vaughan, Mean values of multiplicative functions. Period. Math.Hungar. 43 (2001), no. 1-2, 199–214.[14] H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory I: Classical Theory ,Cambridge Studies in Advanced Mathematics, Vol. 97. Cambridge University Press, 2006.[15] H.L. Montgomery and R.C. Vaughan, Exponential Sums with Multiplicative Functions. In-vent. Math. 43 (1977), 69–82.[16] R. E. A. C. Paley, A theorem on characters , J. London Math. Soc. 7 (1932), 28–32.[17] G. Tenenbaum, Introduction to analytic and probabilistic number theory. Cambridge Studiesin Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995. xvi+448 pp. Department of Mathematics and Statistics, York University, 4700 Keele Street,Toronto, ON, M3J1P3 Canada E-mail address : [email protected] Department of Mathematics, University of Toronto, Toronto, Ontario, Canada E-mail address ::