Related papers: Nonzero values of Dirichlet $L$-functions in verti…

We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $\pi_{q,a}(x)$. Here $\pi_{q,a}(x)$ is…

Let $\chi$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, \chi)$ denote its associated Dirichlet $L$-function. We show that $|L(1, \chi)| \geq 1/(9.12255 \log(q/\pi))$ for sufficiently large $q$, and…

In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions…

We prove a result on the large deviations of the central values of even primitive Dirichlet $L$-functions with a given modulus. For $V\sim \alpha\log\log q$ with $0<\alpha<1$, we show that \begin{equation}\nonumber\frac{1}{\varphi(q)} \#…

The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new.…

We introduce an infinite family of approximations for a Dirichlet $L$-function $L(s, \chi)$ arising from truncated Euler products. These approximations are entire functions and satisfy the same functional equation as $L(s, \chi)$. We…

We show that if $F(s)$ is a nondegenerate ordinary Dirichlet series with nonnegative coefficients and $F(k)$ is a rational number for all large enough positive integers $k$, then the denominators of those rational numbers are unbounded. In…

We consider the non-trivial zeros of the Riemann $\zeta$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line…

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp…

The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy-Littlewood conjecture for pairs of primes in arithmetic…

We show that a positive proportion of the values $L(1/2,\chi_c)$ are non-zero, where $\chi_c$ is the $\ell^{\text{th}}$ residue symbol for $\ell \geq 3$ over $\mathbb{F}_q[t]$, when averaging over square-free polynomials $c$ in…

Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\epsilon> 0$, and $q^{\epsilon} \le N \le q^{1/2 -\epsilon}$, $$ \sum_{\chi \ne \chi_0} |\sum_{n=1}^N \chi(n)|^2 |L(1, \chi)|^2 = (1 + o(1)) \alpha_q q N…

We consider a general form of L-function L(s) defined by an Euler product and satisfies several analytic assumptions. We show several asymptotic formulas for L(1) and log L(1). In particular those asymptotic formulas are valid for Dirichlet…

We investigate special values of Goss L-functions for Dirichlet characters at s=1 over rings of class number one and prove results on their transcendence and algebraic independence.

In this paper, we use the Weyl-bound for Dirichlet $L$-functions to derive zero-density estimates for $L$-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when…

Let $\psi$ be a real primitive character modulo $D$. If the $L$-function $L(s,\psi)$ has a real zero close to $s=1$, known as a Landau-Siegel zero, then we say the character $\psi$ is exceptional. Under the hypothesis that such exceptional…

We prove that, for arbitrary Dirichlet $L$-functions $L(s;\chi_1),\ldots,L(s;\chi_n)$ (including the case when $\chi_j$ is equivalent to $\chi_l$ for $j\ne k$), suitable shifts of type $L(s+i\alpha_jt^{a_j}\log^{b_j}t;\chi_j)$ can…

The aim of this paper is to provide an analogue of the Ball-Rivoal theorem for $p$-adic $L$-values of Dirichlet characters. More precisely, we prove for a Dirichlet character $\chi$ and a number field $K$ the formula…

We establish unconditional sharp upper bounds of the $k$-th moments of the family of quadratic Dirichlet $L$-functions at the central point for $0 \leq k \leq 2$.

We computationally verify that if $L(s,\chi)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(\sigma,\chi) \neq 0$ for real $\sigma \ge 1-1/(5\log q)$. The number of verified moduli exceeds benchmarks due to Watkins…