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## Abstract

We establish the equivalence of conjectures concerning the pair correlation of zeros of L-functions
in the Selberg class and the variances of sums of a related class of
arithmetic functions over primes in short
intervals. This extends the results of Goldston and
Montgomery [‘Pair correlation of zeros and primes in short intervals’,
*Analytic number theory and Diophantine problems (Stillwater, 1984)*, Progress in Mathematics 70 (1987) 183–203] and Montgomery and Soundararajan [‘Primes in short intervals’, *Comm. Math. Phys.* 252 (2004) 589–617] for the Riemann zeta-function to other L-functions
in the Selberg class. Our approach is based on the statistics of the
zeros because the analogue of the Hardy–Littlewood
conjecture for the auto-correlation of the
arithmetic functions we consider is not available in general. One of our
main findings
is that the variances of sums of these arithmetic
functions over primes in short intervals have a different form when the
degree of the associated L-functions
is 2 or higher to that which holds when the degree is 1 (for example,
the Riemann zeta-function). Specifically,
when the degree is 2 or higher, there are two
regimes in which the variances take qualitatively different forms,
whilst in
the degree-1 case there is a single regime.

Original language | English |
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Pages (from-to) | 161-185 |

Number of pages | 25 |

Journal | Journal of the London Mathematical Society |

Volume | 94 |

Issue number | 1 |

Early online date | 1 Jun 2016 |

DOIs | |

Publication status | Published - 1 Aug 2016 |

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