Number Theory

    

The Nontrivial Zeroes of the Riemann Zeta Function are Trivially Expressed by The Euler Product

Authors: Ryan Hackbarth

Leonard Euler considered the Zeta function for real values input of s, and in doing so he famously derived the Euler Product formulation of the Zeta function, which linked the function to prime numbers. The connection between the prime numbers and the Zeta function was further pursued by Bernhard Riemann, who constructed an analytic continuation of the function which remained valid for the entire complex plane. In doing so, he was able to link the zeroes of the Zeta function to the distribution of prime numbers. In this paper, I show that the zeroes of the Zeta Function can be identified directly from the Euler Product, and in doing so, suggest a trivial link between the primes and the zeroes of the Zeta function while bypassing the more complex machinery of analytic continuation.

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[v1] 2026-05-21 23:59:00

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