On the nontrivial zeros of the Dirichlet eta function

Abstract

We construct a two-parameter complex function $η_{κν}:\mathbb{C}\to \mathbb{C}$, $κ\in (0, \infty)$, $ν\in (0,\infty)$ 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 $η_{κν}$ converges to the Dirichlet eta function $η(s)$ as $κ\to \infty$. We prove the crucial property that, for sufficiently large $κ$, the function $η_{κν}(s)$ can be expressed as a linear combination $η_{κν}(s)=\sum_{n=0}^{\infty}a_n(κ) η(s+2νn)$ of horizontal shifts of the eta function (where $a_{n}(κ) \in \mathbb{R}$ and $a_{0}=1$) and that, indeed, we have the inverse formula $η(s)=\sum_{n=0}^{\infty}b_n(κ) η_{κν}(s+2νn)$ as well (where the coefficients $b_{n}(κ) \in \mathbb{R}$ are obtained from the $a_{n}$'s recursively). By using these results and the functional relationship of the eta function, $η(s)=λ(s)η(1-s)$, we sketch a proof of the Riemann hypothesis which, in our setting, is equivalent to the fact that the nontrivial zeros $s^{*}=σ^{*}+it^{*}$ of $η(s)$ (i.e. those points for which $η(s^{*})=η(1-s^{*})=0)$ are all located on the critical line $σ^{*}=\frac{1}{2}$.