2024 The Riemann Hypothesis via the generalizedvon Mangoldt function
William Banks, Saloni Sinha
Funct. Approx. Comment. Math. Advance Publication 1-18 (2024). DOI: 10.7169/facm/240713-22-7

Abstract

Gonek, Graham, and Lee showed the Riemann Hypothesis (RH) can be reformulatedin terms of estimates for twisted sums $\sum_{n\le x}\Lambda(n) n^{-iy}$,where $\Lambda$ is the von Mangoldt function. Using an approachbased on the Perron formula but with a modified contour, we strengthentheir results. We also establish two generalizations,where $\Lambda$ in the twisted sumsis replaced by the generalized von Mangoldt function $$\Lambda_k(n):=\sum_{d\mid n}\mu(d)\Big(\log\frac{n}{d}\Big)^k$$ or the $k$-fold convolution $$\Lambda^k:=\mathop{\underbrace{\Lambda\star\cdots\star\Lambda}}\limits_{k\text{~copies}}.$$ Our results show that the validity of RH is governed by not only the distribution of prime numbers, but also by the distributionof almost-primes, i.e., natural numbers that have no more than$k$ distinct prime divisors.

Citation

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William Banks. Saloni Sinha. "The Riemann Hypothesis via the generalizedvon Mangoldt function." Funct. Approx. Comment. Math. Advance Publication 1 - 18, 2024. https://doi.org/10.7169/facm/240713-22-7

Information

Published: 2024
First available in Project Euclid: 16 December 2024

Digital Object Identifier: 10.7169/facm/240713-22-7

Subjects:
Primary: 11M26
Secondary: 11M06

Keywords: Riemann hypothesis , von Mangoldt function

Rights: Copyright © 2024 Adam Mickiewicz University

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