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arXiv · 1301.1434

Application of an accurate remainder term in the calculation of Residue Class Distributions

Colin Myerscough
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Abstract

This paper uses concepts introduced by Fiorilli and Martin to develop a more accurate remainder term in calculations of the distribution of primes in residue classes. Using it in Rubinstein and Sarnak's method estimates the limiting logarithmic frequency of pi(x)>Li(x) as 0.00000026300 with only 5 zeta zeros used explicitly, and a reliable value between 0.000000262996732 and 0.000000262996733 is obtained with fewer than 100 zeros. Accurate results for "prime number races" can usually be obtained with explicit use of only the first zero of each L-function involved, thus bringing out their dependence on those zeros. For extreme deviations, the method of steepest descent can be applied, and the remainder approximated by an explicit formula together with a rapidly convergent series. This gives, for example, 8.649*10^(-12477) for the logarithmic frequency of pi(x)>Li(x)+5*Li(sqrt(x)). The Monach - Lamzouri model of the extreme distribution is developed to give close agreement with these results. The remainder can also be calculated explicitly as an asymptotic series. This allows good modelling of non-extreme races and the calculation of distributions by direct convolution. The three different methods agree to within 0.001% where they overlap in application. The distribution calculated from values of pi(x) for x<10^20 shows similar behaviour to the limiting distribution, but is somewhat further from normal.

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Colin Myerscough. 2013-01-08. Application of an accurate remainder term in the calculation of Residue Class Distributions. https://arxiv.org/abs/1301.1434

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