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Publication - Professor Jon Keating

    On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class

    Citation

    Bui, HM, Keating, JP & Smith, DJ, 2016, ‘On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class’. Journal of the London Mathematical Society, vol 94., pp. 161-185

    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.

    Full details in the University publications repository