A Reformulation of the Riemann Hypothesis

TL;DR

The paper targets the non-trivial zeros of the Riemann zeta function by constructing a pole-free, entire companion function $\varphi(k)$ whose zeros align with the zeta zeros in the critical strip. It builds a right-half-plane analytic continuation of $\zeta(k)$ through the Dirichlet eta function and a transformed polylogarithm framework, yielding an explicit integral representation for $\varphi(k)$ and a direct relation to $\zeta(k)$ via $\varphi(k)=-\dfrac{2\Gamma(k+1)\left(2^{1-k}-1\right)}{\pi^k}\cos{\dfrac{\pi k}{2}}\,\zeta(k)$; the key reformulation asserts that RH is equivalent to the zeros of $\varphi(k)$ lying on the critical line $\Re(k)=\tfrac{1}{2}$. A further set of results includes a simple integral expression for $\varphi(k)$, an integer-valued analysis at specific $k$, a generating function $q(x)$ with $\varphi(k)=\dfrac{q^{(k)}(x)}{k!}$, and a graphical methodology (via plots of $f(r,t)$ and $g(r,t)$) to visualize the zero structure. Together, these contribute a new analytic framework for studying RH and offer tools for qualitative and numerical investigation of zeta zeros.

Abstract

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to the right half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $\varphi(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.