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Computing the Mertens function on a GPU

Eugene Kuznetsov
arXiv:1108.0135v1 [math.NT] (31 Jul 2011)

@article{2011arXiv1108.0135K,

   author={Kuznetsov}, E.},

   title={"{Computing the Mertens function on a GPU}"},

   journal={ArXiv e-prints},

   archivePrefix={"arXiv"},

   eprint={1108.0135},

   primaryClass={"math.NT"},

   keywords={Mathematics – Number Theory},

   year={2011},

   month={jul},

   adsurl={http://adsabs.harvard.edu/abs/2011arXiv1108.0135K},

   adsnote={Provided by the SAO/NASA Astrophysics Data System}

}

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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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