The Riemann Hypothesis via the generalized von Mangoldt Function (2209.11768v2)

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.