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The distribution of consecutive prime biases and sums of sawtooth random variables

Published online by Cambridge University Press:  02 August 2018

ROBERT J. LEMKE OLIVER  and
KANNAN SOUNDARARAJAN
ROBERT J. LEMKE OLIVER
Affiliation:
Department of Mathematics, Tufts University, 503 Boston Ave. Medford, MA 02155, U.S.A. e-mail: [email protected]
KANNAN SOUNDARARAJAN
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380. Stanford, CA 94305-2125, U.S.A. e-mail: [email protected]
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Abstract

In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 149 - 169
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

Partially supported by NSF grant DMS-1601398.

‡

Partially supported by NSF grant DMS-1500237 and by a Simons Investigator grant from the Simons Foundation.

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