An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

Abstract:

In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character $\chi$ to squarefree modulus $q$, we prove the following upper bound: \begin{align*} \left | \sum _{1 \leqslant n\leqslant N} \chi (n) \right |\leqslant c \sqrt {q} \log q, \end{align*} 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.