A theory for the zeros of Riemann $\zeta$ and other $L$-functions (updated)

TL;DR

This paper reviews key properties of the Riemann zeta function, introduces new theoretical results linking zeros to solutions of transcendental equations, and explores strategies towards proving the Riemann Hypothesis, including numerical analysis and generalizations.

Contribution

It presents a novel approach connecting zeros of the zeta function to solutions of a transcendental equation, offering a new perspective on the Riemann Hypothesis and its generalizations.

Findings

Zeros on the critical line correspond to zeros of a cosine function

A transcendental equation for the zeros depends only on their index n

Numerical analysis supports the theoretical framework and explains failures in known counterexamples

Abstract

In these lectures we first review the important properties of the Riemann $\zeta$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic continuation, the functional equation, trivial zeros, the Euler product formula, Riemann's main result relating the zeros on the critical strip to the distribution of primes, the exact counting formula for the number of zeros on the strip $N(T)$, and the GUE statistics of the zeros on the critical line. We then turn to presenting some new results obtained in the past year and describe several strategies towards proving the RH. First we describe an electrostatic analogy and argue that if the electric potential along the line $\Re (z) =1$ is a regular alternating function, the RH would follow. The main new result is that the zeros on the critical line are in one-to-one correspondence with the zeros of the cosine function, and this leads to a transcendental equation for the $n$-th zero on the critical line that depends only on $n$. If there is a unique solution to this equation for every $n$, denoting $N_0 (T)$ the number of zeros on the critical line, then $N_0(T) = N(T)$, i.e. all zeros are on the critical line. These results are generalized to two infinite classes of functions, Dirichlet $L$-functions and $L$-functions based on modular forms. We present extensive numerical analysis of the solutions of these equations. We apply these methods to the Davenport-Heilbronn $L$-function, which is known to have zeros off of the line, and explain why the RH fails in this case. We also present a new approximation to the $\zeta$-function that is analogous to the Stirling approximation to the $\Gamma$-function. In this updated version we added references to subsequent publications that made progress on open questions posed in the previous version from 10 years ago.