An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

: Matteo Bordignon, Bryce Kerr

Publisher: AMER MATHEMATICAL SOC

: 2020

: Transactions of the American Mathematical Society

: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

: T AM MATH SOC

: 373

: 9

: 6503

: 6527

: 25

: 0002-9947

: 1088-6850

DOI: https://doi.org/10.1090/tran/8138

In this paper we obtain a new fully explicit constant for the Polya-Vinogradov inequality for squarefree modulus. Given a primitive character chi to squarefree modulus q, we prove the following upper bound:vertical bar Sigma(1 <= n <= N) chi(n)vertical bar <= c root q log q,where c = 1/(2 pi(2)) + o(1) for even characters and c = 1/(4 pi) + o(1) for odd characters, with an explicit o(1) term. This improves a result of Frolenkov and Soundararajan for large q. We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of log q as in previous approaches and is an important factor for fully explicit bounds.