Geodesic
A Pólya–Vinogradov inequality for short character sums
Canadian mathematical bulletin, Tome 64 (2021) no. 4, pp. 906-910

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DOI

In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$, with $0\le \gamma \le 1/3$. We prove that $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.
DOI : 10.4153/S0008439520000934
Mots-clés : Character sums, Pólya-Vinogradov inequality 
Bordignon, Matteo. A Pólya–Vinogradov inequality for short character sums. Canadian mathematical bulletin, Tome 64 (2021) no. 4, pp. 906-910. doi: 10.4153/S0008439520000934
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     title = {A {P\'olya{\textendash}Vinogradov} inequality for short character sums},
     journal = {Canadian mathematical bulletin},
     pages = {906--910},
     year = {2021},
     volume = {64},
     number = {4},
     doi = {10.4153/S0008439520000934},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000934/}
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