Related papers: Effective uniform approximation by $L$-functions i…

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of…

Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $|z|\leq r$, that is analytic in the interior. Voronin's universality theorem asserts that translates of the Riemann zeta function $\zeta(3/4 + z + it)$ can approximate $f$…

We prove explicit formulae for $\alpha$-points of $L$-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function $\zeta(s)$ to the case of $\alpha$-points of…

We survey the results and the methods in the theory of universality for various zeta and $L$-functions, obtained in these forty years after the first discovery of the universality for the Riemann zeta-function by Voronin.

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…

In 1975 Voronin proved the universality theorem for the Riemann zeta-function $\zeta(s)$ which roughly says that any admissible function $f(s)$ is approximated by $\zeta(s)$. A few years later Reich proved a discrete analogue of this…

In 2003, Garunk\v{s}tis provided a lower bound for the lower density of the universality theorem for the Riemann zeta-function. In this paper, we generalize this result for the hybrid joint universality theorem for Dirichlet $L$-functions…

We prove an effective universality theorem of the Riemann zeta-function in short intervals $[T,T+H]$ with $T^{\frac{27}{82}}\le H\le T$ by following an effective multidimensional $\Omega$-result of Voronin. Furthermore, we also prove the…

In 2017, Garunk\v{s}tis, Laurin\v{c}ikas and Macaitien\.{e} proved the discrete universality theorem for the Riemann zeta-function sifted by the nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended in…

We establish the universality theorem for the first four symmetric power L-functions of automorphic forms and their associated Rankin-Selberg L-functions. This generalizes some results of Laurincikas & Matsumoto and Matsumoto respectively.

We prove that the Voronin universality theorem for the Riemann zeta-function extends to the line Re(s)=1 if in addition to vertical shifts we also allow scaling and adding a sufficiently large constant.

We prove the Voronin universality theorem for the multiple Hurwitz zeta-function with rational or transcendental parameters in $\mathbb{C}^n$ answering a question of Matsumoto. In particular this implies that the Euler-Zagier multiple…

We prove the universality theorem for the iterated integrals of logarithms of $L$-functions in the Selberg class on some line parallel to the real axis.

It is well known that the Riemann zeta function, as well as several other $L$-functions, is universal in the strip $1/2<\sigma<1$; this is certainly not true for $\sigma>1$. Answering a question of Bombieri and Ghosh, we give a simple…

Laurin\v{c}ikas proposed that whether the universality theorem for the Riemann zeta-function shifted by an exponential function holds or not. This problem is solved for more general shifts by Andersson et al. In this paper, we generalize…

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and…

This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some…

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist--Meurman--type congruences for the universal Bernoulli polynomials that are related…

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem…

This article deals with applications of Voronin's universality theorem for the Riemann zeta-function $\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values…