Publication

Some applications of the Weil-Barner explicit formula

Thorsten Kleinjung
2013
Journal Articles

Abstract

The Weil-Barner explicit formula is applied to the prime counting function and to the problem of numerical partial verification of the Riemann hypothesis.

Official source

https://infoscience.epfl.ch/entities/publication/0dcef1ac-6b88-4ba0-b511-2d16f3a58135

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Thorsten Kleinjung
2013
Journal Articles

Abstract

The Weil-Barner explicit formula is applied to the prime counting function and to the problem of numerical partial verification of the Riemann hypothesis.

Official source

https://infoscience.epfl.ch/entities/publication/0dcef1ac-6b88-4ba0-b511-2d16f3a58135

About this result

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Related concepts (16)

Prime-counting function

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by pi(x) (unrelated to the number pi). Prime number theorem Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately where log is the natural logarithm, in the sense that This statement is the prime number theorem.

Riemann hypothesis

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named.

Prime omega function

In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby (little omega) counts each distinct prime factor, whereas the related function (big omega) counts the total number of prime factors of honoring their multiplicity (see arithmetic function). That is, if we have a prime factorization of of the form for distinct primes (), then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.

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