Related papers: Explicit upper bound for $\left|L(1, \chi)\right|$…

Let $F$ be a linear combination of $N\geq 1$ Dirichlet $L$-functions attached to even (or odd) primitive characters with the same modulus. Selberg proved that a positive proportion of non-trivial zeros of $F$ lie on the critical line. Our…

We obtain an asymptotic formula for the first moment of quadratic Dirichlet $L$--functions over function fields at the central point $s=\tfrac{1}{2}$. Specifically, we compute the expected value of $L(\tfrac{1}{2},\chi)$ for an ensemble of…

Let $f\in S_k(N,\psi)$ be a newform, and let $\chi$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is a prime and $\ell>1$. In this paper we describe a method to establish convexity breaking bounds of the form $$…

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…

Let $(f_1,f_2)$ a $2$-tuple of arithmetic functions and $$\Delta_3(n,f_1,f_2):=\sup\limits_{\substack{(u_1,u_2) \in \mathbb{R}^{2} \\(v_1,v_2) \in [0,1]^{2}}}\Big\lvert \sum\limits_{\substack{d_1 d_{2} \mid n \\ e^{u_i}<d_i\leqslant…

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime…

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^\eta$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes\chi)$ when $f$ is a primitive…

Let $L(s,\chi)$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\Re(s)=1/2$, we show that $\gg T \log T$ of them are not zeros of $L(s,\chi)$. This result provides some theoretical…

For an irreducible orthogonal character $\chi $ of even degree there is a unique square class $\det({\chi })$ in the character field such that the invariant quadratic forms in any $L$-representation affording $\chi $ have determinant in…

For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is…

In this paper, we prove the upper and lower bounds for normal derivatives of spectral clusters $u=\chi_{\lambda}^s f$ of Dirichlet Laplacian $\Delta_M$, $$c_s \lambda\|u\|_{L^2(M)} \leq \| \partial_{\nu}u \|_{L^2(\partial M)} \leq C_s…

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group $G$ of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a…

Given a rational elliptic curve $ E $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ \chi $ of order $ q $, and in many cases its associated central algebraic L-value $ \mathcal{L}(E, \chi) $…

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for…

In this article, we study the logarithm of the central value $L\left(\frac{1}{2}, \chi_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $\delta$ over a fixed finite field…

For any irreducible character $\chi$ of a finite group $G$, let $\theta(\chi)$ denote the proportion of elements $g\in G$ for which $\chi(g)$ is either zero or a root of unity. Then for any $L\in[1/2,1]$ and any $\epsilon>0$, there exists…

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k},$$ where…

For a finite group $G$ and an irreducible complex character $\chi$ of $G$, the codegree of $\chi$ is defined by $\textrm{cod}(\chi)=|G:\textrm{ker}(\chi)|/\chi(1)$, where $\textrm{ker}(\chi)$ is the kernel of $\chi$. In this paper, we show…

For any $\theta>\frac13$, we show that there are constants $c_1,c_2>0$ that depend only on $\theta$ for which the following property holds. If $\chi_1,\chi_2$ are two distinct primitive Dirichlet characters modulo $q$, and $T\ge…

For a character $\chi$ of a finite group $G$, the number cod$(\chi):=|G:\mathrm{ker}(\chi)|/\chi(1)$ is called the codegree of $\chi$.In this paper, we give a solvability criterion for a finite group $G$ depending on the minimum of the…