On the nontrivial zeros of the Dirichlet eta function

TL;DR

This paper introduces a new complex function that approximates the Dirichlet eta function and uses it to sketch a proof that all nontrivial zeros lie on the critical line, offering a novel approach to the Riemann hypothesis.

Contribution

The paper constructs a two-parameter complex embedding of the eta function and demonstrates its use in providing a new proof sketch of the Riemann hypothesis.

Findings

Constructed a holomorphic nonlinear embedding of the eta function.

Showed the embedding can be expressed as a linear combination of shifted eta functions.

Provided a sketch of a proof that nontrivial zeros are on the critical line.

Abstract

We construct a two-parameter complex function $\eta_{\kappa \nu}:\mathbb{C}\to \mathbb{C}$, $\kappa \in (0, \infty)$, $\nu\in (0,\infty)$ that we call a holomorphic nonlinear embedding and that is given by a double series which is absolutely and uniformly convergent on compact sets in the entire complex plane. The function $\eta_{\kappa \nu}$ converges to the Dirichlet eta function $\eta(s)$ as $\kappa \to \infty$. We prove the crucial property that, for sufficiently large $\kappa$, the function $\eta_{\kappa \nu}(s)$ can be expressed as a linear combination $\eta_{\kappa \nu}(s)=\sum_{n=0}^{\infty}a_n(\kappa) \eta(s+2\nu n)$ of horizontal shifts of the eta function (where $a_{n}(\kappa) \in \mathbb{R}$ and $a_{0}=1$) and that, indeed, we have the inverse formula $\eta(s)=\sum_{n=0}^{\infty}b_n(\kappa) \eta_{\kappa \nu}(s+2\nu n)$ as well (where the coefficients $b_{n}(\kappa) \in \mathbb{R}$ are obtained from the $a_{n}$'s recursively). By using these results and the functional relationship of the eta function, $\eta(s)=\lambda(s)\eta(1-s)$, we sketch a proof of the Riemann hypothesis which, in our setting, is equivalent to the fact that the nontrivial zeros $s^{*}=\sigma^{*}+it^{*}$ of $\eta(s)$ (i.e. those points for which $\eta(s^{*})=\eta(1-s^{*})=0)$ are all located on the critical line $\sigma^{*}=\frac{1}{2}$.