Preprint Article Version 1 This version is not peer-reviewed

# The Zeros of the Dirichlet Beta Function Encode the Odd Primes and Have Real Part 1/2

Version 1 : Received: 19 April 2018 / Approved: 24 April 2018 / Online: 24 April 2018 (05:10:20 CEST)

How to cite: Lander, A. The Zeros of the Dirichlet Beta Function Encode the Odd Primes and Have Real Part 1/2. Preprints 2018, 2018040305 (doi: 10.20944/preprints201804.0305.v1). Lander, A. The Zeros of the Dirichlet Beta Function Encode the Odd Primes and Have Real Part 1/2. Preprints 2018, 2018040305 (doi: 10.20944/preprints201804.0305.v1).

## Abstract

It is well known that the primes and prime powers have a deep relationship with the nontrivial zeros of Riemann’s zeta function. This is a reciprocal relationship. The zeros and the primes are encoded in each other and are reciprocally recoverable. Riemann’s zeta is an extended or continued version of Euler’s zeta function which in turn equates with Euler’s product formula over the primes. This paper shows that the zeros of the converging Dirichlet or Catalan beta function, which requires no continuation to be valid in the critical strip, can be easily determined. The imaginary parts of these zeros have a deep and reciprocal relationship with the odd primes and odd prime powers. This relationship separates the odd primes into those having either 1 or 3 as a remainder after division by 4. The vector pathway of the beta function is such that the real part of its zeros has to be a half.

## Subject Areas

Catalan beta function; Riemann’s zeta function; primes; Dirichlet L-function