Some notes about the Zeta function and the Riemann hypothesis

TL;DR

This paper explores an algebraic extension of the Zeta Function to analyze the Riemann Hypothesis, suggesting that zero-points must lie on the critical line due to symmetry and compatibility conditions.

Contribution

It introduces a novel algebraic approach using an extended Zeta Function based on Euler-McLaurin integration to examine zero distribution.

Findings

Zero-points must lie on the critical line due to symmetry constraints.

Compatibility conditions imply zeros are constrained to the critical line.

The approach offers new algebraic insights into the Riemann Hypothesis.

Abstract

The present essay aims at investigating whether and how far an algebraic analysis of the Zeta Function and of the Riemann Hypothesis can be carried out. Of course the well-established properties of the Zeta Function, explored in depth in over 150 years of world-wide study, are taken for granted. The chosen approach starts from the recognized necessity of formulating an extension of the Zeta Function which is defined for Re(s) = X < 1. A particular form of extension, based on a Euler-McLaurin integration procedure, is chosen and an algebraic appraisal is made of the conditions required to make the expression of this extension equal to zero. Imposing the zero-condition on this extension of the Zeta Function implies the nullity of the sum of three terms with different properties. This can be obtained only if certain compatibility conditions between the three terms are satisfied. One of the terms is endowed with a high degree of symmetry with respect to the Critical Line (C.L.) X = 1/2, and from the compatibility conditions it follows that either the other two terms possess the same type of symmetry or any complex-conjugate (c.c.) pair of zero-points should lie on the C.L. Since the symmetry properties of the first term are not shared by the other two terms, it appears that the only allowed c.c. pairs of zero-points must belong to the C.L.