Related papers: Joint universality for dependent $L$-functions

We prove an explicit upper bound of the function $S(t,\chi)$, defined by the argument of Dirichlet $L$-functions. An explicit upper bound of the function $S_1(t)$, defined by the integral of the argument of the Riemann zeta-function, have…

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…

Double $L$-functions are the generalization of Dirichlet $L$-functions to two variable functions. We investigate the order estimation of double $L$-functions, and give upper bounds which are explicit in conductor aspect.

Approximation in measure is employed to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann Hypothesis are…

In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation,…

Let $U\subseteq\mathbb{R}^{n}$ be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function $f:U\to\mathbb{R}$ can be approximated by real analytic convex functions, uniformly on all of $U$. In doing so we…

In the context of the correspondence between real functions on the unit circle and inner analytic functions within the open unit disk, that was presented in previous papers, we show that the constructions used to establish that…

In 2023, Li, Du, Yi proved a uniqueness theorem for L functions in the extended Selberg class under the assumptions of positive degree, a shared functional equation, and the sharing of three complex values. This was later strengthened by…

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical…

The main purpose of this paper is to establish bounds on the second moment of $L\big(\tfrac{1}{2}+it,\chi\big)$, averaged over families of fixed order characters. A discrete version of the main result is also stated, from which zero-density…

The purpose of this paper is to generalize our earlier work on the logarithm of the Riemann zeta-function to linear combinations of logarithms of primitive Dirichlet $L$-functions with constant real coefficients. Under the assumption of…

We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $\pi(x;q,a)<\pi(x;q,b)$ for a set of real $x$ of…

We prove Local Central Limit Theorems (LLT) for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is a Markov chains with equicontinuous conditional probabilities satisfying contraction…

The property of zeta-functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with accuracy $\epsilon>0$ by suitable vertical shifts of the pair consisting…

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet $L$--functions over function fields. More specifically, we study the average of $L(1/2+\alpha,\chi_D)/L(1/2+\beta,\chi_D)$, when $D$ varies over monic,…

Let $\xi_0,\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|\xi_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} \xi_k f_{k,n} z^k, $$ where…

In this paper we deal with the task of uniformly approximating an $L$-biLipschitz curve by means of piecewise linear ones. This is rather simple if one is satisfied to have approximating functions which are $L'$-biLipschitz, for instance…

We prove a lower bound of the correct order of magnitude in the conductor aspect for small rational moments of Drichlet $L$-functions. Such bounds require new techniques, which is visible from the relationship to non-vanishing results for…

We obtain uniform lower bounds, true for all automorphic L-functions L(s) associated to cuspidal representations of GL(m,A) where A denotes the adeles of the rationals Q, of the integral on the vertical line (Re(s)=1/2) of the absolute…

We establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet $L$-function of fixed modulus under the generalized Riemann hypothesis.