Dirichlet Character Research Articles

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.

Towards the Generalized Riemann Hypothesis using only zeros of the Riemann zeta function

For any real $$\beta _0\in [\tfrac{1}{2},1)$$ , let $$\textrm{GRH}[\beta _0]$$ be the assertion that for every Dirichlet character $$\chi $$ and all zeros $$\rho =\beta +i\gamma $$ of $$L(s,\chi )$$ , one has $$\beta \leqslant \beta _0$$ (in particular, $$\textrm{GRH}[\frac{1}{2}]$$ is the Generalized Riemann Hypothesis). In this paper, we show that the validity of $$\textrm{GRH}[\frac{9}{10}]$$ depends only on certain distributional properties of the zeros of the Riemann zeta function $$\zeta (s)$$ . No conditions are imposed on the zeros of nonprincipal Dirichlet L-functions.

Jacobi forms, Saito-Kurokawa lifts, their Pullbacks and sup-norms on average

We formulate a precise conjecture about the size of the $$L^\infty $$ -mass of the space of Jacobi forms on $$\mathbb {H}_n \times \mathbb C^{g \times n}$$ of matrix index S of size g. This $$L^\infty $$ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when $$n=1$$ , $$g \ge 1$$ . When $$n=1$$ and $$g=1$$ , we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger’s formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values $$L(1/2, f \otimes \chi _D)$$ with f varying over newforms of level a prime p and even weight k as $$k,p \rightarrow \infty $$ and D being (explicitly) polynomially bounded by k, p. Here $$\chi _D$$ is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is $$k^{5/2}$$ by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded $$\mathbb {H}\times \mathbb {H}$$ is $$k^2$$ . In an appendix, the same question is answered for the pullbacks of the whole space $$S^2_k$$ , the size here being $$k^3$$ .

The Second Moment Theory of Families of 𝐿-Functions–The Case of Twisted Hecke 𝐿-Functions

For a fairly general family of L L -functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q q , and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values L ( f ⊗ χ , 1 / 2 ) L(f\otimes \chi ,1/2) are non-zero, and indeed bounded from below; there exist many characters χ \chi for which the central L L -value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.

Mean square values of L-functions over subgroups for nonprimitive characters, Dedekind sums and bounds on relative class numbers

AbstractAn explicit formula for the mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$ , provided that $d\ll \frac {\log p}{\log \log p}$ . Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$ , we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$ . As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ . Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

A unified strategy to compute some special functions of number-theoretic interest

We introduce an algorithm to compute the functions belonging to a suitable set F defined as follows: f∈F means that f(s,x), s∈A⊂R being fixed and x>0, has a power series expansion centred at x0=1 with convergence radius greater or equal than 1; moreover, it satisfies a functional equation of step 1 and the Euler-Maclaurin summation formula can be applied to f. Denoting the Euler gamma-function as Γ, we will show that, for x>0, log⁡Γ(x), the digamma function ψ(x), the polygamma functions ψ(w)(x), w∈N, w≥1, and, for s>1 being fixed, the Hurwitz ζ(s,x)-function and its first partial derivative ∂ζ∂s(s,x) are in F. In all these cases the coefficients of the involved power series will depend on the values of ζ(u), u>1, where ζ is the Riemann zeta-function. As a by-product, we will also show how to compute the Dirichlet L-functions L(s,χ) and L′(s,χ), s>1, χ being a primitive Dirichlet character, by inserting the reflection formulae of ζ(s,x) and ∂ζ∂s(s,x) into the first step of the Fast Fourier Transform algorithm. Moreover, we will obtain some new formulae and algorithms for the Dirichlet β-function and for the Catalan constant G. Finally, we will study the case of the Bateman G-function and of the alternating Hurwitz zeta-function, also known as the η-function; we will show that, even if they are not in F, our approach can be adapted to handle them too. In the last section we will also describe some tests that show a performance gain with respect to a standard multiprecision implementation of ζ(s,x) and ∂ζ∂s(s,x), s>1, x>0.

On the generalized Cochrane sum with Dirichlet characters

<abstract><p>In this paper, we defined a new generalized Cochrane sum with Dirichlet characters, and gave the upper bound of the generalized Cochrane sum with Dirichlet characters. Moreover, we studied the asymptotic estimation problem of the mean value of the generalized Cochrane sum with Dirichlet characters and obtained a sharp asymptotic formula for it. By using this asymptotic formula, we also gave the mean value of the generalized Dedekind sum.</p></abstract>

Moments of traces of Frobenius of higher order Dirichlet $L$-functions over $\mathbb F_q[T

We study the moments of ${\rm Tr}(\Theta _\chi)$ as $\chi $ runs over Dirichlet characters defined over $\mathbb F_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behaves lik

On the Mean Value of High-Powers of a Special Character Sum Modulo a Prime

In this paper, we use the elementary methods, the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime, and derive an exact computational formula. It can be conveniently programmed by the “Mathematica” software, by which we can get the exact results easily.

Correlations of multiplicative functions in function fields

AbstractWe develop an approach to study character sums, weighted by a multiplicative function , of the form where χ is a Dirichlet character and ξ is a short interval character over . We then deduce versions of the Matomäki–Radziwiłł theorem and Tao's two‐point logarithmic Elliott conjecture over function fields , where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin–Shusterman on correlations of the Möbius function for various values of q. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that q is a power of 2. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the existing one in the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a “corrected” form of the Erdős discrepancy problem over .

Weyl-type bounds for twisted GL(2) short character sums

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for \(SL(2,{\mathbb {Z}})\) with normalized Fourier coefficients \(\lambda _{f}(n)\). Let \(\chi \) be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL(2) short character sum. Here, we consider the problem of cancellation in short character sum of the form $$\begin{aligned} S_{f,\chi }(N):= \mathop \sum _{n \in {\mathbb {Z}}}\lambda _{f}(n)\chi (n)W\Big (\frac{n}{N}\Big ). \end{aligned}$$We show that, for \(0<\theta < \frac{1}{10}\), $$\begin{aligned} S_{f,\chi }(N) \ll _{f,\epsilon }N^{3/4 + \theta /2}p^{1/6}(pN)^{\epsilon } + N^{1-\theta }(pN)^{\epsilon }, \end{aligned}$$which is non-trivial if \(N \ge p^{2/3 + \alpha + \epsilon }\) where\(\alpha = = \frac{4\theta }{1-6\theta }\). Previously, such a bound was known for \(N \ge p^{3/4 + \epsilon }.\)

On linear independence of Dirichlet L values

The study of linear independence of L(k,χ) for a fixed integer k>1 and varying χ depends critically on the parity of k vis-à-vis χ. This has been investigated by a number of authors for Dirichlet characters χ of a fixed modulus and having the same parity as k. The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a single number field lurking in the background). This entails a very careful and hands-on dealing with the arithmetic of compositum of number fields which we undertake in this work. Our results extend earlier works of the first author with Murty-Rath as well as works of Okada, Murty-Saradha and Hamahata.

Two general series identities involving modified Bessel functions and a class of arithmetical functions

AbstractWe consider two sequences $a(n)$ and $b(n)$ , $1\leq n&lt;\infty $ , generated by Dirichlet series $$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$ satisfying a familiar functional equation involving the gamma function $\Gamma (s)$ . Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$ , and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$ , the Bessel functions of imaginary argument $I_{\mu }(z)$ , and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$ . Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$ , the number of representations of n as a sum of k squares $r_k(n)$ , and primitive Dirichlet characters $\chi (n)$ .

On the vanishing of twisted L-functions of elliptic curves over rational function fields

We investigate in this paper the vanishing at $$s=1$$ of the twisted L-functions of elliptic curves E defined over the rational function field $${\mathbb {F}}_q(t)$$ (where $${\mathbb {F}}_q$$ is a finite field of q elements and characteristic $$\ge 5$$ ) for twists by Dirichlet characters of prime order $$\ell \ge 3$$ , from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li (J Number Theory 191:85–103, 2018) and Donepudi and Li (Rocky Mountain J Math 51(5):1615–1628, 2021) who proved vanishing at $$s=1/2$$ for infinitely many Dirichlet L-functions over $${\mathbb {F}}_q(t)$$ based on the existence of one, and we can prove that if there is one $$\chi _0$$ such that $$L(E, \chi _0, 1)=0$$ , then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over $${\mathbb {F}}_q(t)$$ behave differently than the general ones.

On an extension of a question of Baker

It is an open question of Baker whether the numbers [Formula: see text] for nontrivial Dirichlet characters [Formula: see text] with period [Formula: see text] are linearly independent over [Formula: see text]. The best known result is due to Baker, Birch and Wirsing which affirms this when [Formula: see text] is co-prime to [Formula: see text]. In this paper, we extend their result to any arbitrary family of moduli. More precisely, for a positive integer [Formula: see text], let [Formula: see text] denote the set of all [Formula: see text] values as [Formula: see text] varies over nontrivial Dirichlet characters with period [Formula: see text]. Then for any finite set of pairwise co-prime natural numbers [Formula: see text] with [Formula: see text] [Formula: see text], we show that the set [Formula: see text] is linearly independent over [Formula: see text]. In the process, we also extend a result of Okada about linear independence of the cotangent values over [Formula: see text] as well as a result of Murty–Murty about [Formula: see text] linear independence of such [Formula: see text] values. Finally, we prove [Formula: see text] linear independence of such [Formula: see text] values of Erdösian functions with distinct prime periods [Formula: see text] for [Formula: see text] with [Formula: see text] [Formula: see text].

On the Chowla and twin primes conjectures over $\mathbb{F}_q[T

Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb{F}_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions and resolve Chowla's $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Möbius function, we obtain a level of distribution beyond $1/2$ for irreducible polynomials, and establish the twin prime conjecture in a quantitative form. All these results hold for finite fields satisfying a simple condition.

Joint approximation by non-linear shifts of Dirichlet L-functions

In the paper, a theorem on the simultaneous approximation of a collection of analytic functions by non-linear shifts of Dirichlet L-functions (L(s+itτα1,χ1),…,L(s+itταr,χr)) is obtained. Here tτ is the Gram function, α1,…,αr are fixed different positive numbers, and χ1,…,χr are arbitrary Dirichlet characters. Also, an example of approximation by a certain composition of the above shifts is given.

On a generalization of Menon–Sury identity to number fields involving a Dirichlet character

For every positive integer n, Sita Ramaiah’s identity states that $$\begin{aligned}&\sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} \gcd (a_1+a_2-1,n) = \phi _2(n)\sigma _0(n) \\&\quad \text { where } \; \phi _2(n)= \sum _{a_1, a_2, a_1+a_2 \in (\mathbb {Z}/n\mathbb {Z})^*} 1, \end{aligned}$$where \((\mathbb {Z}/n\mathbb {Z})^*\) is the multiplicative group of units of the ring \(\mathbb {Z}/n\mathbb {Z}\) and \(\sigma _s(n) = \displaystyle \sum \nolimits _{d\mid n}d^s\). This identity can also be viewed as a generalization of Menon’s identity. In this article, we generalize this identity to an algebraic number field K involving a Dirichlet character \(\chi \). Our result is a further generalization of recent results in Ji and Wang (Ramanujan J 53:585–594, 2020) and Sury (Rend Circ Mat Palermo 58:99–108, 2009).

On the Conditional Bounds for Siegel Zeros

In this paper, under a weakened version of Hardy—Littlewood Conjecture on the number of representations in Goldbach problem, we shall prove bounds for the Siegel zeros of real primitive Dirichlet characters for composite moduli.

One-level density of the family of twists of an elliptic curve over function fields

We fix an elliptic curve E/Fq(t) and consider the family {E⊗χD} of E twisted by quadratic Dirichlet characters. The one-level density of their L-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside (−1,1). As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least 12.5% of the family have rank zero, and at least 37.5% have rank one. The Katz and Sarnak philosophy predicts that those percentages should both be 50% and that the average analytic rank should be 1/2. We finish by computing the one-level density of E twisted by Dirichlet characters of order ℓ≠2 coprime to q. We obtain a restriction of (−1/2,1/2) on the support with a unitary symmetry.