The Riemann Hypothesis via the generalized 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.