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

@inproceedings{Steuding2009AspectsOA, title={Aspects of Analytic Number Theory : The Universality of the Riemann Zeta-Function}, author={J{\"o}rn Steuding}, year={2009} }

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.

#### One Citation

Disjoint universality connected with differential operators

- Mathematics
- 2020

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 from… Expand

#### References

SHOWING 1-10 OF 55 REFERENCES

The Theory of the Riemann Zeta-Function

- Mathematics
- 1987

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 aspects… Expand

An introduction to the value-distribution theory of zeta-functions

- Mathematics
- 2006

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 the… Expand

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 on… Expand

Limit Theorems for the Riemann Zeta-Function

- Mathematics
- 1995

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 Riemann… Expand

The effective universality theorem for the Riemann zeta function

- 2003

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

- Mathematics
- 2007

Dirichlet Series and Polynomial Euler Products.- Interlude: Results from Probability Theory.- Limit Theorems.- Universality.- The Selberg Class.- Value-Distribution in the Complex Plane.- The Riemann… Expand

On the effectivization of the universality theorem for the Lerch zeta-function

- Mathematics
- 2000

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 the… Expand

The joint universality of twisted automorphic $L$-functions

- Mathematics
- 2004

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

- Mathematics
- 2006

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 note… Expand

Topics in Multiplicative Number Theory

- Mathematics
- 1971

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 value… Expand