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A generalization of the Riemann–Siegel formula

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Abstract

The celebrated Riemann–Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable t. Siegel anticipated that this formula could be generalized to include the Hardy–Littlewood approximate functional equation, valid in any vertical strip. We give this generalization for the first time. The asymptotics contain Mordell integrals and an interesting new family of polynomials.

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  1. Department of Mathematics, The CUNY Graduate Center, 365 Fifth Avenue, New York, NY, 10016-4309, USA

    Cormac O’Sullivan

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O’Sullivan, C. A generalization of the Riemann–Siegel formula. Math. Z. 303, 20 (2023). https://doi.org/10.1007/s00209-022-03164-8

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