Dirichlet Character Research Articles

Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8

Abstract In this article, we compute binomial convolution sums of divisor functions associated with the Dirichlet character modulo 8, which is the remaining primitive Dirichlet character modulo powers of 2 yet to be considered. To do this, we provide an explicit expression of a general modular form of even weight for Γ 0 ( 32 ) {\Gamma }_{0}\left(32) in terms of its basis, and use an identity of Huard, Ou, Spearman, and Williams. As an application, we also compute the number of representations by quadratic forms under parity conditions.

Eigenspaces of newforms with nontrivial character

Let [Formula: see text] denote the space of holomorphic cuspforms with Dirichlet character [Formula: see text] and modular subgroup [Formula: see text]. We will characterize the space of newforms [Formula: see text] as the intersection of eigenspaces of a particular family of Hecke operators, generalizing previous work of Baruch–Purkait to forms with nontrivial character. We achieve this by obtaining representation theoretic results in the [Formula: see text]-adic case which we then de-adelize into relations of classical Hecke operators.

Non-zero central values of Dirichlet twists of elliptic L-functions

We consider heuristic predictions for small non-zero algebraic central values of twists of the L-function of an elliptic curve E/Q by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem associated to E/Q extended to chosen families of cyclic extensions of fixed degree.

The Generalized Riemann Hypothesis from zeros of a single [formula omitted]-function

For each primitive Dirichlet character χ, a hypothesis GRH†[χ] is formulated in terms of zeros of the associated L-function L(s,χ). It is shown that for any such character, GRH†[χ] is equivalent to the Generalized Riemann Hypothesis.

On the Paley graph of a quadratic character

Abstract Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293–306] about the size of this upper bound.

On Some Multipliers Related to Discrete Fractional Integrals

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓp→ℓq bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators.

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS

Dense clusters of zeros near the zero-free region of ζ(s)

The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function [Formula: see text] of the form [Formula: see text] For many decades, the general shape of the zero-free region has not changed (although explicit known values for [Formula: see text] have improved over the years). In this paper, we show that if the zero-free region cannot be widened substantially, then there exist infinitely many distinct dense clusters of zeros of [Formula: see text] lying close to the edge of the zero-free region. Our proof provides specific information about the location of these clusters and the number of zeros contained in them. To prove the result, we introduce and apply a variant of the original method of de la Vallée Poussin combined with ideas of Turán to control the real parts of power sums. We also prove similar results for [Formula: see text]-functions associated to nonquadratic Dirichlet characters [Formula: see text] modulo [Formula: see text].

Magic squares, the symmetric group and Möbius randomness

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the k-fold divisor function.

Reciprocity formulae for generalized Dedekind–Rademacher sums attached to three Dirichlet characters and related polynomial reciprocity formulae

We define a three-character analogue of the generalized Dedekind–Rademacher sum introduced by Hall, Wilson, and Zagier and prove its reciprocity formula which contains all of the reciprocity formulas in the literature for generalized Dedekind–Rademacher sums attached (and not attached) to Dirichlet characters as special cases. Additionally, we prove related polynomial reciprocity formulas which contain all of the polynomial reciprocity formulas in the literature as special cases, such as those given by Carlitz, Beck & Kohl, and the present author.

Extreme values of L$L$‐functions of newforms

AbstractIn 2008, Soundararajan showed that there exists a normalized Hecke eigenform of weight and level one such that for sufficiently large . In this note, we show that for any and for all sufficiently large , the number of normalized Hecke eigenforms of weight and level one for which is . For an odd fundamental discriminant , let be the set of all cuspidal normalized Hecke eigenforms of weight and level dividing . When the real primitive Dirichlet character satisfies , we investigate the number of for which takes extremal values.

The ratios conjecture for real Dirichlet characters and multiple Dirichlet series

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros. We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis. This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives near the critical line. As an application, we compute the one-level density for test functions whose Fourier transform is supported in ( − 2 , 2 ) \left (-2,2\right ) , including lower-order terms.

A Study of Some Problems on the Dirichlet Characters (mod q)

Consider q to be an integer, and χ to be a Dirichlet character (mod q). The main purpose of this paper is using the analytic methods to study some problems related to the Dirichlet character (mod q).

One kind two-term exponential sums weighted by third-order character

<abstract><p>We give a correction to the previous paper of S. F. Cao and T. T. Wang (2021). Additionally, provide a conversion formula of the power mean of a certain exponential sums weighted by a Dirichlet character $ \chi_{3} $ and some of its applications.</p></abstract>

The fourth power mean of the generalized quadratic Gauss sums associated with some Dirichlet characters

<abstract><p>In this paper, the fourth power mean values of the generalized quadratic Gauss sums associated with the $ 3 $-order and $ 4 $-order Dirichlet characters are given by using the properties of the Dirichlet characters and Gauss sums.</p></abstract>

A generalization of T\'oth identity involving a Dirichlet character in $\mathbb{F}_q[T

A generalization of T\'oth identity involving a Dirichlet character in $\mathbb{F}_q[T

Average value of the divisor class numbers of real cubic function fields

Abstract We compute an asymptotic formula for the divisor class numbers of real cubic function fields K m = k ( m 3 ) {K}_{m}=k\left(\sqrt[3]{m}) , where F q {{\mathbb{F}}}_{q} is a finite field with q q elements, q ≡ 1 ( mod 3 ) q\equiv 1\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) , k ≔ F q ( T ) k:= {{\mathbb{F}}}_{q}\left(T) is the rational function field, and m ∈ F q [ T ] m\in {{\mathbb{F}}}_{q}\left[T] is a cube-free polynomial; in this case, the degree of m m is divisible by 3. For computation of our asymptotic formula, we find the average value of ∣ L ( s , χ ) ∣ 2 {| L\left(s,\chi )| }^{2} evaluated at s = 1 s=1 when χ \chi goes through the primitive cubic even Dirichlet characters of F q [ T ] {{\mathbb{F}}}_{q}\left[T] , where L ( s , χ ) L\left(s,\chi ) is the associated Dirichlet L L -function.

Mollified moments of cubic Dirichlet L-functions over the Eisenstein field

We prove, assuming the generalized Riemann Hypothesis (GRH) that there is a positive density of L-functions associated with primitive cubic Dirichlet characters over the Eisenstein field that do not vanish at the central point s=1/2. This is achieved by computing the first mollified moment, which is obtained unconditionally, and finding a sharp upper bound for the higher mollified moments for these L-functions, under GRH. The proportion of non-vanishing is explicit, but extremely small.

Hybrid subconvexity bounds for twists of GL(3) L-functions

Let [Formula: see text] be a Hecke–Maass cusp form on [Formula: see text] and [Formula: see text] a primitive Dirichlet character of prime power conductor [Formula: see text] with [Formula: see text] prime. In this paper, we will prove the following subconvexity bound [Formula: see text] for any [Formula: see text] and [Formula: see text].

Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$\mathbb {Z}_n$$

Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$\mathbb {Z}_n$$