On the nontrivial zeros of the Dirichlet eta function

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 $s^{\ast }={\sigma }^{\ast }+i{t}^{\ast }$ of $\eta (s)$ (i.e. those points for which $\eta (s^{\ast })=\eta (1-s^{\ast })=0)$ are all located on the critical line ${\sigma }^{\ast }=\frac{1}{2}$.

17 pages, submitted for publication

References in corpus (3)

• From deterministic cellular automata to coupled map lattices
• Supersymmetry and the Riemann zeros on the critical line
• Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding