Related papers: Modified Dirichlet character sums over the $k$-fre…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq…

The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \left( \frac{1}{2},\chi\right)$ are non-vanishing as $\chi$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond…

In this paper, we establish sharp upper bounds on shifted moments of the family of Dirichlet $L$-functions to a fixed modulus over function fields. We apply the result to obtain upper bounds on moments of Dirichlet character sums over…

We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the…

We obtain (conditional and unconditional) results on large values of $L$-functions $L(s,\chi)$ in the critical strip $1/2 \leq \Re s \leq 1$ when the character $\chi$ runs through a thin subgroup of all characters modulo an integer $q$.…

We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for…

We show that the truncated simplex Hilbert transform enjoys some cancellation in the sense that its norm grows sublinearly in the number of scales retained in the truncation. This extends the recent result by Tao on cancellation for the…

Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been…

Answering a question of Cameron, Pretzel and Siemons proved that every integer partition of $n\ge 2(k+3)(k+1)$ can be reconstructed from its set of $k$-deletions. We describe a new reconstruction algorithm that lowers this bound to $n\ge…

We prove cancellation in a sum of Fourier coefficents of a GL(3) form $F$ twisted by additive characters, uniformly in the form $F$.

We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…

We show under the Generalised Riemann Hypothesis that for every $\delta>0$, almost every prime $q$ in $[Q,2Q]$ has the expected of prime primitive roots in the interval $[x,x+x^{\frac{1}2+\delta}]$ provided $Q$ is not more than…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet…

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…

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring $\text{Int}(D)=\{f\in K[x]\mid f(D)\subseteq D\}$,…

The Fourier transform is approximated over a finite domain using a Riemann sum. This Riemann sum is then expressed in terms of the discrete Fourier transform, which allows the sum to be computed with a fast Fourier transform algorithm more…

We develop a formalism for performing real space renormalization group transformations of the "decimation type" using perturbation theory. The type of transformations beyond $d=1$ is nontrivial even for free theories. We check the formalism…

In 2016, Cramer, Ducas, Peikert and, Regev proposed an efficient algorithm for recovering short generators of principal ideals in $q$-th cyclotomic fields with $q$ being a prime power. In this paper, we improve their analysis of the dual…