The Riemann Hypothesis via the generalized von Mangoldt Function

TL;DR

This paper explores new equivalences of the Riemann Hypothesis using twisted sums with generalized von Mangoldt functions, linking RH to the distribution of almost-primes and providing explicit asymptotic estimates.

Contribution

It extends previous reformulations of RH by analyzing twisted sums with generalized von Mangoldt functions for all natural numbers k, establishing new equivalences.

Findings

RH is equivalent to specific asymptotic estimates for twisted sums with Λ_k.

Explicit formulas relate RH to the distribution of almost-primes.

Similar results are obtained for the k-fold convolution of the von Mangoldt function.

Abstract

Gonek, Graham, and Lee have shown recently that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with von Mangoldt function $\Lambda$. Building on their ideas, for each $k\in\mathbb{N}$, we study twisted sums with the \emph{generalized von Mangoldt function} $$ \Lambda_k(n):=\sum_{d\,\mid\,n}\mu(d)\Big(\log\frac{n}{d}\,\Big)^k $$ and establish similar connections with RH. For example, for $k=2$ we show that RH is equivalent to the assertion that, for any fixed $\epsilon>0$, the estimate $$ \sum_{n\leq x}\Lambda_2(n)n^{-iy} =\frac{2x^{1-iy}(\log x-C_0)}{(1-iy)} -\frac{2x^{1-iy}}{(1-iy)^2} +O\big(x^{1/2}(x+|y|)^\epsilon\big) $$ holds uniformly for all $x,y\in\mathbb{R}$, $x\geq 2$; hence, the validity of RH is governed by the distribution of almost-primes in the integers. We obtain similar results for the function $$ \Lambda^k:=\mathop{\underbrace{\,\Lambda\star\cdots\star\Lambda\,}}\limits_{k\text{~copies}}\,, $$ the $k$-fold convolution of the von Mangoldt function.