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Mean Values of Cubic and Quartic Dirichlet Characters 3 MEAN VALUES OF CUBIC AND QUARTIC DIRICHLET CHARACTERS PENG GAO AND LIANGYI ZHAO Abstract. We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan [3] on quadratic Dirichlet character sums. Mathematics Subject Classification (2010): 11L05, 11L40 Keywords: cubic Dirichlet character, quartic Dirichlet character 1. Introduction Estimations for character sums have wide applications in number theory. A fundamental result is the well-known P´olya-Vinogradov inequality (see for example [4, Chap. 23]), which asserts that for any non-principal Dirichlet character χ modulo q, M Z and N N, ∈ ∈ (1.1) χ(n) q1/2 log q. ≪ M<n M+N X≤ One may regard the P´olya-Vinogradov inequality as a mean value estimate for characters and one expects to obtain asymptotic formulas if an extra average over the characters is introduced. We are thus led to the investigation on the following expression χ(n) χ S n X∈ X where S is a certain set of characters. For example, when S is the set of all Dirichlet characters modulo q, then it follows from the orthogonality relation of the characters that the above sum really amounts to a counting on the number of integers which are congruent to 1 modulo q. Another interesting choice for S is to take S to contain characters of a fixed order. A basic and important case is the set of quadratic characters. In [3], J. B. Conrey, D. W. Farmer and K. Soundararajan studied the following sum: m S (X, Y )= , 2 n m X n Y (m,X≤2)=1 (n,X≤2)=1 m where n is the Jacobi symbol. arXiv:1902.10987v3 [math.NT] 16 Jan 2020 While it is relatively easy to obtain an asymptotic formula of S2(X, Y ) when Y = o(X/ log X) or X = o(Y/ log Y ) using (1.1), it is more subtle to treat the remaining XY -ranges. Using a Poisson summation formula developed in [17], a valid asymptotic formula of S2(X, Y ) for all X, Y is obtained in [3]. Most interestingly, the formula exhibits a transition in the behavior of S2(X, Y ) when X and Y are of comparable size. Recently, the authors [7] studied the mean values of some quadratic, cubic and quartic Hecke characters. These are analogues to S2(X, Y ) in number fields. Again, the most interesting case is when X, Y are of comparable size. Another similar behavior is shown to exist by I. Petrow in an earlier study [16] on the mean values of shifted convolution sums of Fourier coefficients of Hecke eigenforms. In this paper, we return to the classical setting by studying the mean values of cubic and quartic Dirichlet characters. To form such sums, we note that primitive cubic Dirichlet characters exist modulo any rational prime p if and only if p 1 (mod 3) and there are precisely two such characters which are complex conjugate of each other when they exist. The≡ same conclusions apply to primitive quartic Dirichlet characters as well except in we need to have p 1 (mod 4). Based on these observations, we introduce the following sets of Dirichlet characters that we aim to study.≡ We define Date: January 17, 2020. 1 2 PENG GAO AND LIANGYI ZHAO the set S3,1 to be the set that contains the principal Dirichlet character modulo 1. For any integer n> 1, we define the set S3,n to be non-empty if and only if n is a product of powers of primes which are congruent to 1 modulo 3, in which case by writing n = k pαi with p 1 (mod 3) and α 1, we define i=1 i i ≡ i ≥ Q k αi (1.2) S3,n = χi : χi primitive cubic Dirichlet character modulo pi . (i=1 ) Y Similarly, let S4,1 be the set that contains the principal Dirichlet character modulo 1. For any integer n > 1, we define the set S4,n to be non-empty if and only if n is a product of powers of primes which are congruent to 1 modulo 4, in which case by writing n = k pαi with p 1 (mod 4) and α 1, we define i=1 i i ≡ i ≥ Qk αi (1.3) S4,n = χi : χi primitive quartic Dirichlet character modulo pi . (i=1 ) Y We are now ready to define the following character sums of our interest. For i = 3, 4, X, Y > 1, let Si(X, Y )= χ(m). n Y χ Si,n m X X≤ ∈X X≤ Our goal in this paper is to evaluate Si(X, Y ) asymptotically for i =3, 4. Our method here only allows us to treat the situation in which Y X, a condition we shall assume henceforth. In this case, one expects that the main contribution ≤ for Si(X, Y ) comes from the terms when n is a cube if i = 3 or a fourth-power if i = 4. Treating the remaining terms using the P´olya-Vinogradov inequality (1.1), we deduce that 1/i XY 3/2+ǫ (1.4) Si(X, Y )= Ci + O Y , i =3, 4, √log Y for some constants Ci. The above allows us to obtain asymptotic formulas for Si(X, Y ),i = 3, 4 when Y is small compared to X. Thus, it is again a subtlety to obtain asymptotic formulas for Si(X, Y ),i =3, 4 when Y is close to X. This is precisely what we want to study in this paper. In view of (1.4), we may assume that Y X6/7 when studying ≥ S (X, Y ) and that Y X4/5 when studying S (X, Y ). Our main result is 3 ≥ 4 Theorem 1.1. For large X Y and any ε> 0, we have for Y X6/7, ≥ ≥ 1/3 1/3 5/17 XY XY 4/3 24/17+ǫ Y (1.5) S3(X, Y )= C1 + O + Y + Y . √log Y (log Y )3/2 X ! We also have for Y X4/5, ≥ 1/4 1/4 9/31 38/31 XY XY 91/62+ǫ Y 13/31+ǫ Y (1.6) S4(X, Y )= C2 + O + Y + XY . √log Y (log Y )3/2 X X ! Here C1, C2 are constants given in (3.13) and (3.23), respectively. A simple computation will show that (1.5) is a valid asymptotic formula (the main term dominates all the O-terms) 33/35 δ 160/187 δ if Y = O(X − ) and (1.6) if Y = O(X − ) for any δ > 0. We point out here that C1 and C2 are positive (see the discussions below (3.13) and (3.23)). Our proof of Theorem 1.1 follows the line of treatment in [3] by first applying the Poisson summation formula to covert the sum over m in Si(X, Y ) to its dual sum. The zero frequency gives us the main term as usual. To treat the contribution of remaining terms, we transform the sum over n in Si(X, Y ), using Lemma 2.2, into another sum over algebraic integers in an imaginary quadratic number field. The resulting sums involve with certain Gauss sums for which we shall use a result of S. J. Patterson [15] (see Lemma 2.5). This treatment is also inspired by the method used in [1]. 1.2. Notations. The following notations and conventions are used throughout the paper. e(z) = exp(2πiz)= e2πiz, ω = e(1/3). f = O(g) or f g means f cg for some unspecified positive constant c. ε denotes an arbitrary≪ small| |≤ positive number, which may be different from line to line. N(n) denotes the norm of n Z[ω] or n Z[i]. ̟ denotes a prime element in∈Z[ω] or Z[i∈] and p denotes a prime in Z. MEAN VALUES OF CUBIC AND QUARTIC DIRICHLET CHARACTERS 3 2. Preliminaries 2.1. Cubic and quartic symbols. For any number field K, let K denote the ring of integers in K and UK the group of units in . Throughout this paper, set K = Q(ω),K = Q(Oi), where ω = exp(2πi/3). It is well-known that both OK ω i Q(i) and Q(ω) have class number one and that Kω = Z[ω], Ki = Z[i]. Recall that every ideal in Z[ω] co-prime to 3 has a unique generator congruent to 1 modulo 3O (see [2, PropositionO 8.1.4]) and every ideal in Z[i] coprime to 2 has a unique generator congruent to 1 modulo (1 + i)3 (see the paragraph above Lemma 8.2.1 in [2])). These generators are called primary. Let n· 3 be the cubic residue symbol in Kω . For a prime ̟ Kω with N(̟) = 3, the cubic symbol is defined for a (N(̟) 1)/O3 a ∈ O 2 6 a a , (a,̟)=1 by a − (mod ̟), with 1,ω,ω . When ̟ a, we define = 0. Then ∈ OKω ̟ 3 ≡ ̟ 3 ∈{ } | ̟ 3 the cubic symbol can be extended to any composite n with (N(n), 3) = 1 multiplicatively. We extend the definition of · to n U by setting · = 1. Recall that [13, Theorem 7.8] the cubic reciprocity law states that for two n 3 ∈ Kω n 3 primary m,n Kω , ∈ O m n = . n 3 m 3 The quartic case is similar.
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