Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus

Alfred Wünsche

doi:10.4236/apm.2016.613074

Abstract

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions with an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions , possesses zeros only on the imaginary axis. The Riemann zeta function as it is known can be related to an entire function with the same non-trivial zeros as . Then after a trivial argument displacement we relate it to a function with a representation of the form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. Besides this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.

Highlights

The Riemann zeta function ζ (s) which basically was known already to Euler establishes the most important link between number theory and analysis
The Riemann hypothesis is the conjecture that all nontrivial zeros of the Riemann zeta function ζ ( s) for complex s= σ + it are positioned on the line s= 1 + it that means on the line parallel to the imaginary axis through real value 2 σ = 1 in the complex plane and in extension that all zeros are simple zeros [2]-[17] 2 (with extensive lists of references in some of the cited sources, e.g., ([4] [5] [9] [12] [14])
We proved in this article the Riemann hypothesis embedded into a more general theorem for a class of functions Ξ ( z ) with a representation of the form (3.1) for realvalued functions Ω (u ) which are positive semi-definite and non-increasing in the interval 0 ≤ u < +∞ and which are vanishing in infinity more rapidly than any exponential function exp (−λu) with λ > 0

Summary

Introduction

The Riemann zeta function ζ (s) which basically was known already to Euler establishes the most important link between number theory and analysis. To the Xi function in mentioned integral transform we apply the second mean-value theorem of real analysis first on the imaginary axes and discuss its extension from the imaginary axis to the whole complex plane For this purpose we derive in Appendix. The Riemann hypothesis for the zeta function ζ (s= σ + it ) is equivalent to the hypothesis that all zeros of the related entire function Ξ ( z =x + iy) lie on the imaginary axis z = iy that means on the line to real part x = 0 of z= x + iy which becomes the critical line. We will go another way where the restriction to these strips does not play a role for the proof

Application of Second Mean-Value Theorem of Calculus to Xi Function

Specialization of Second Mean-Value Theorem to Xi Function on Imaginary Axis

Accomplishment of Proof for Zeros of Xi Functions on Imaginary Axis Alone

Some Consequences from Proof of the Riemann Hypothesis

Equivalent Formulations of the Main Theorems in a Summary

Conclusion