Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters
In this paper we obtain a new fully explicit constant for the Pólya–Vinogradov inequality for primitive characters. Given a primitive character \chi modulo q , we prove the following upper bound: \Big| \sum_{1 \le n\le N} \chi(n) \Big|\le c \sqrt{q} \log q, where c=3/(4\pi^2)+o_q(1) for even charact...
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| Published in | Revista matemática iberoamericana Vol. 38; no. 4; pp. 1101 - 1127 |
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| Main Author | |
| Format | Journal Article |
| Language | English Spanish |
| Published |
European Mathematical Society Publishing House
01.06.2022
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| Online Access | Get full text |
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| Summary: | In this paper we obtain a new fully explicit constant for the Pólya–Vinogradov inequality for primitive characters. Given a primitive character
\chi
modulo
q
, we prove the following upper bound:
\Big| \sum_{1 \le n\le N} \chi(n) \Big|\le c \sqrt{q} \log q,
where
c=3/(4\pi^2)+o_q(1)
for even characters and
c=3/(8\pi)+o_q(1)
for odd characters, with explicit
o_q(1)
terms. This improves a result of Frolenkov and Soundararajan for large
q
. We proceed, following Hildebrand, to obtain the explicit version of a result by Montgomery–Vaughan on partial Gaussian sums and an explicit Burgess-like result on convoluted Dirichlet characters. |
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| ISSN: | 0213-2230 2235-0616 |
| DOI: | 10.4171/rmi/1328 |