Computing the Mertens function on a GPU

Eugene Kuznetsov

Computing the Mertens function on a GPU

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arxiv 1108.0135 v1 pith:AYYIIJ6Q submitted 2011-07-31 math.NT

Eugene Kuznetsov This is my paper · ORCID

classification math.NT

keywords algorithmapproxfunctionmertensapproximatecomputecomputingdiscussed

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abstract

A GPU implementation of an algorithm to compute the Mertens function in O(x2/3+{\ko}) time is discussed. Results for x up to $10^{22}$, and a new extreme value for $M(x)/x^{1/2}$, -0.585768 ($M(x) \approx -1.996 \ast 10^9$ at $x \approx 1.161 \ast 10^{19}$), are reported.An approximate algorithm is used to examine values of M(x) for x up to $\exp{(10^{15})}$.

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Practical Computations of the Mertens Function: $M(10^{24})$ and $M(10^{25})$
math.NT 2026-07 accept novelty 4.0 of 10

M(10^24) = 7,189,337,839 and M(10^25) = -258,560,632,948 are computed using an optimized O(x^{2/3+epsilon}) algorithm, extending the record by two orders of magnitude.

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