Corpus ID: 4133703

Aspects of Analytic Number Theory : The Universality of the Riemann Zeta-Function

  title={Aspects of Analytic Number Theory : The Universality of the Riemann Zeta-Function},
  author={J{\"o}rn Steuding},
Abstract. These notes deal with Voronin’s universality theorem which states, roughly speaking, that any non-vanishing analytic function can be uniformly approximated by certain shifts of the Riemann zeta-function. We start with a brief introduction to the classical theory of the zeta-function. Then we give a self-contained proof of the universality theorem. We conclude with several interesting applications of this remarkable property and discuss some related problems and extensions. 

Figures from this paper

Disjoint universality connected with differential operators
In this article, we study disjoint universality for certain sequences of operators, that are connected with the differential operator. Actually, the motivation to study such sequences comes fromExpand


The Theory of the Riemann Zeta-Function
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspectsExpand
An introduction to the value-distribution theory of zeta-functions
This is an expository survey on the value-distribution theory of zeta-functions, mainly of the Riemann zeta-function. We discuss denseness results, the universality, and limit theorems both in theExpand
Probabilistic value-distribution theory of zeta-functions
The starting point of the value-distribution theory of zeta-functions is Bohr’s achievement in the first half of the 20th century, who proved the denseness results and probabilistic limit theorems onExpand
Limit Theorems for the Riemann Zeta-Function
Preface. 1. Elements of the probability theory. 2. Dirichlet series and Dirichlet polynomials. 3. Limit theorems for the modulus of the Riemann Zeta-function. 4. Limit theorems for the RiemannExpand
The effective universality theorem for the Riemann zeta function
It is known that Voronin’s universality theorem for the Riemann zetafunction is ineffective. For some partial cases we obtain the effective version of this theorem.
Value-Distribution of L-Functions
Dirichlet Series and Polynomial Euler Products.- Interlude: Results from Probability Theory.- Limit Theorems.- Universality.- The Selberg Class.- Value-Distribution in the Complex Plane.- The RiemannExpand
On the effectivization of the universality theorem for the Lerch zeta-function
In the paper one way of the effectivization of the universality theorem for the Lerch zeta-function is discussed. It is based on the estimate of the rate of convergence in a limit theorem in theExpand
The joint universality of twisted automorphic $L$-functions
The simultaneous universality of twisted automorphic L-functions, associated with a new form with respect to a congruence subgroup of SL(2; Z) and twisted by Dirichlet characters, is proved.Expand
On the Value Distribution of Shifts of Universal Dirichlet Series
Abstract.Shifts of universal Dirichlet series are jointly universal, that means, roughly speaking, they approximate simultaneously any given family of non-vanishing analytic functions. In this noteExpand
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean valueExpand