Related papers: Numerical Computations Concerning Landau-Siegel Ze…

We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of…

We prove that $|\zeta(\sigma+it)|\le 70.7 |t|^{4.438 (1-\sigma)^{3/2}}\log^{2/3}|t|$ for $1/2\le\sigma\le 1$ and $|t|\ge 3$. As a consequence, we improve the explicit zero-free region for $\zeta(s)$, showing that $\zeta(\sigma+it)$ has no…

Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then the Hurwitz-Lerch zeta function is defined by $\Phi (s,a,z) := \sum_{n=0}^\infty z^n(n+a)^{-s}$ when $\sigma :=\Re (s) >1$. In this paper, we show that the Hurwitz zeta…

Let $\chi$ be an idele class character over a number field $F$, and let $\pi,\pi'$ be any two cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that the Rankin-Selberg $L$-function…

We continue the investigation into the computational status of the existence of moduli of regularity (and their use for rates of convergence) in the sense of Kohlenbach, Lopez and Nicolae (2019), carried out w.r.t. classical reverse…

We prove that among 1 and the odd zeta values $\zeta(3)$, $\zeta(5)$, \ldots, $\zeta(s)$, at least $ 0.21 \sqrt{s}/\sqrt{\log s}$ are linearly independent over the rationals, for any sufficiently large odd integer $s$. This is the first…

Y\i ld\i r\i m has classified zeros of the derivatives of Dirichlet $L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this paper we remove the possibility of vagrant zeros for $L'(s,\chi)$ when the conductors are…

Ingham (1940) proved that $N(\sigma,T)\ll T^{3(1-\sigma)/(2-\sigma)}\log^{5}{T}$, where $N(\sigma,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}\geq\sigma\geq 1/2$ and $0<\Im\{\rho\}\leq…

Let $d$ be a real number, let $s$ be in a fixed compact set of the strip $1/2<\sigma<1$, and let $L(s, \chi)$ be the Dirichlet $L$-function. The hypothesis is that for any real number $d$ there exist 'many' real numbers $\tau$ such that the…

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the…

In this paper, we study the number of additional zeros of Dirichlet $L$-function caused by multiplicity by using Asymptotic Large Sieve. Then in asymptotic terms we prove that there are more than 80.124% of zeros of the family of Dirichlet…

An explicit hybrid estimate for $L(1/2+it,\chi)$ is derived, where $\chi$ is a Dirichlet character modulo $q$. The estimate applies when $t$ is bounded away from zero, and is most effective when $q$ is powerfull, yielding an explicit Weyl…

We study the relation between the size of $L(1,\chi)$ and the width of the zero-free interval to the left of that point.

This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L'(\sigma;\chi)$ for $\sigma$ close to unity, using a result on the…

Let $E(s, Q)$ be the Epstein zeta function attached to a positive definite quadratic form of discriminant $D<0$, such that $h(D)\geq 2$, where $h(D)$ is the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{D})$. We denote by…

We improve on previous upper bounds for the $q$th norm of the partial sums of the Riemann zeta function on the half line when $0<q\leqslant 1$. In particular, we show that the 1-norm is bounded above by $(\log N)^{1/4}(\log\log N)^{1/4}$.

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<\delta _0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with…

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=\Div B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with complex-valued bounded…

Let $f$ be a fixed (holomorphic or Maass) modular cusp form. Let $\cq$ be a Dirichlet character mod $q$. We describe a fast algorithm that computes the value $L(1/2,f\times\chi_q)$ up to any specified precision. In the case when $q$ is…

In recent years a variant of the resonance method was developed which allowed to obtain improved $\Omega$-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be…