Euler Product Sieve

TL;DR

This paper explores Euler product-based approximations of the Riemann zeta function, justifies the Euler Product Sieve for primes, and links the Bounded Riemann Hypothesis to the Riemann Hypothesis and zero simplicity.

Contribution

It introduces a new class of approximations to the zeta function based on Euler products and connects the Bounded Riemann Hypothesis to the classical Riemann Hypothesis plus zero simplicity.

Findings

Euler Product Sieve is justified for prime generation

New approximations to the zeta function are proposed

Bounded Riemann Hypothesis is equivalent to Riemann Hypothesis plus zero simplicity

Abstract

We study a class of approximations to the Riemann zeta function introduced earlier by the second author on the basis of Euler product. This allows us to justify Euler Product Sieve for generation of prime numbers. Also we show that Bounded Riemann Hypothesis (stated in a paper by the fourth author) is equivalent to conjunction the Riemann Hypothesis + the simplicity of zeros. 16 pages.