A multiple sum involving the Möbius function.

Motohashi, Yoichi

Displaying similar documents to “A multiple sum involving the Möbius function.”

On a problem of Fujii concerning Riemann's $ζ$ -function.

Puchta, J.-C. (2001)

Acta Mathematica Universitatis Comenianae. New Series

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Gaps between zeros of the derivative of the Riemann $ξ$ -function

Hung Manh Bui (2010)

Journal de Théorie des Nombres de Bordeaux

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Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of $ξ^{'} (s)$ . We prove that a positive proportion of gaps are less than $0.796$ times the average spacing and, in the other direction, a positive proportion of gaps are greater than $1.18$ times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than $0.7203$ ( $1.5$ , respectively).

On Robin’s criterion for the Riemann hypothesis

YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Solé (2007)

Journal de Théorie des Nombres de Bordeaux

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Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality $σ (n) : = \sum_{d | n} d < e^{γ} n log log n$ is satisfied for $n \geq 5041$ , where $γ$ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if $n \geq 37$ does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that $n$ must be divisible by a fifth power $> 1$ . As consequence we obtain that RH holds true iff every natural number divisible by...