Polya-Vinogradov best constant (squarefree asymptotic)

Description of constant

Let $\chi$ be a primitive Dirichlet character modulo $q$, and define

\[S(\chi)\ :=\ \max_{N\le q}\ \left\lvert\sum_{1\le n\le N}\chi(n)\right\rvert.\]

The Polya-Vinogradov inequality states that

\[S(\chi)\ \le\ c\,\sqrt{q}\,\log q\]

for some absolute constant $c$.

[BK2020-def-PV]

For squarefree moduli, define $C_{72}^{\mathrm{even}}$ (resp. $C_{72}^{\mathrm{odd}}$) as the infimum of all $c$ such that

\[S(\chi)\ \le\ (c+o(1))\,\sqrt{q}\,\log q \quad\text{as }q\to\infty\]

for every primitive even (resp. odd) character $\chi$ modulo squarefree $q$. Define

\[C_{72}\ :=\ \max\{C_{72}^{\mathrm{even}},\,C_{72}^{\mathrm{odd}}\}.\]

Bordignon and Kerr proved, for squarefree $q$, that one can take

\[c=\frac{1}{2\pi^2}+o(1)\ \text{(even)},\qquad c=\frac{1}{4\pi}+o(1)\ \text{(odd)}.\]

[BK2020-main-constants]

Hence

\[C_{72}\ \le\ \frac{1}{4\pi}.\]

Known upper bounds

Bound	Reference	Comments
$\dfrac{2}{\pi^2}$	[Pom2011]	Primitive characters, explicit inequality; asymptotically this gives $c=2/\pi^2$ (even) and $c=1/(2\pi)$ (odd), so $C_{72}\le 2/\pi^2$. [Pom2011-thm1]
$\dfrac{3}{8\pi}$	[B2022]	All primitive moduli (hence also squarefree), odd characters; gives $C_{72}^{\mathrm{odd}}\le 3/(8\pi)$ and thus $C_{72}\le 3/(8\pi)$. [B2022-main]
$\dfrac{1}{\pi^2}$	[Kerr2020]	Cubefree moduli (hence squarefree), arbitrary intervals; implies the same leading constant for the initial-interval quantity $S(\chi)$. [Kerr2020-main]
$\dfrac{1}{4\pi}$	[BK2020]	Squarefree moduli, odd characters; this controls $C_{72}=\max{C_{72}^{\mathrm{even}},C_{72}^{\mathrm{odd}}}$. [BK2020-main-constants]
$\dfrac{1}{2\pi^2}$	[BK2020]	Squarefree moduli, even characters. [BK2020-main-constants]

Known lower bounds

Bound	Reference	Comments
$0$		Trivial from nonnegativity of the defining infimum.

Additional comments and links

The best known asymptotic leading constants differ between even and odd characters in the squarefree setting. [BK2020-main-constants]
BK2020 identifies Frolenkov-Soundararajan as the previous sharpest explicit benchmark and improves it (for large $q$) in the squarefree setting. [BK2020-prev-best] [FS2013]
A follow-up by Bordignon gives fully explicit constants for all primitive moduli, namely $3/(4\pi^2)+o_q(1)$ (even) and $3/(8\pi)+o_q(1)$ (odd), improving Frolenkov-Soundararajan for large $q$. [B2022-main]
Kerr obtains a cubefree-modulus bound with leading constant $1/\pi^2$ for arbitrary intervals; this is weaker than BK2020 in the squarefree setting but still relevant context. [Kerr2020-main]
Wikipedia page on the Polya-Vinogradov inequality

References

[BK2020] Bordignon, Matteo; Kerr, Bryce. An explicit Polya-Vinogradov inequality via Partial Gaussian sums. Transactions of the American Mathematical Society 373 (2020), no. 9, 6503-6527. DOI: https://doi.org/10.1090/tran/8138. arXiv PDF: https://arxiv.org/pdf/1909.01052.pdf. Google Scholar
- [BK2020-def-PV] loc: arXiv PDF p.1, Introduction, definition of $S(\chi)$ and displayed inequality (1) quote: “Given two integers $N$, $q$ and a primitive character $\chi$ modulo $q$ consider the sums $S(\chi):=\max_{N\le q}\left\lvert\sum_{1\le n\le N}\chi(n)\right\rvert$. A bound, proven independently by Polya and Vinogradov in the early 1900s, is the following $S(\chi)\le c\sqrt{q}\log q$ for some absolute constant $c$.”
- [BK2020-main-constants] loc: arXiv PDF p.1, Abstract quote: “Given a primitive character $\chi$ to squarefree modulus $q$, we prove the following upper bound $\left\lvert \sum_{1 \le n\le N} \chi(n) \right\rvert\le c \sqrt{q} \log q$, 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.”
- [BK2020-prev-best] loc: arXiv PDF p.4, Introduction, paragraph beginning “Fully explicit Pólya-Vinogradov inequalities have previously been considered …” quote: “Fully explicit Pólya-Vinogradov inequalities have previously been considered by Frolenkov [15], Frolenkov and Soundararajan [16] and Pomerance [27]. The current sharpest result is Frolenkov and Soundararajan [16].”
[B2022] Bordignon, Matteo. Partial Gaussian sums and the Pólya-Vinogradov inequality for primitive characters. Revista Matemática Iberoamericana 38 (2022), no. 4, 1101-1127. DOI: https://doi.org/10.4171/RMI/1328. Google Scholar
- [B2022-main] loc: journal PDF p.1101 (first page), Abstract quote: “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 $\left\lvert \sum_{1 \le n\le N} \chi(n) \right\rvert\le c \sqrt{q} \log q$, 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 $q$.”
[Kerr2020] Kerr, Bryce. On the constant in the Pólya-Vinogradov inequality. Journal of Number Theory 212 (2020), 265-284. DOI: https://doi.org/10.1016/j.jnt.2019.11.003. arXiv PDF: https://arxiv.org/pdf/1807.09573.pdf. Google Scholar
- [Kerr2020-main] loc: arXiv source Polya-Vinogradov_-_constant.tex, Section 2 (Main result), Theorem 1 quote: “For integer $q$ we define $c=\begin{cases} \frac{1}{4} \quad \text{if $q$ is cubefree}, \ \frac{1}{3} \quad \text{otherwise}. \end{cases}$ For any primitive character $\chi \mod{q}$ and integers $M$ and $N$ we have $\left|\sum_{M<n< M+N}\chi(n)\right|\le (1+o(1))\frac{4c}{\pi^2}q^{1/2}\log{q}.$”
[Pom2011] Pomerance, Carl. Remarks on the Pólya-Vinogradov Inequality. Integers 11A (2011), Article A19. DOI: https://doi.org/10.1515/integ.2011.039. PDF: https://math.colgate.edu/~integers/a16int2009/a16int2009.pdf. Google Scholar
- [Pom2011-thm1] loc: Integers PDF p.3, Theorem 1 quote: “For $\chi$ a primitive character to the modulus $q>1$, we have $S(\chi)\le \frac{2}{\pi^2}q^{1/2}\log q+\frac{4}{\pi^2}q^{1/2}\log\log q+\frac{3}{2}q^{1/2}$ if $\chi$ is even, and $S(\chi)\le \frac{1}{2\pi}q^{1/2}\log q+\frac{1}{\pi}q^{1/2}\log\log q+q^{1/2}$ if $\chi$ is odd.”
[FS2013] Frolenkov, D. A.; Soundararajan, K. A generalization of the Pólya-Vinogradov inequality. The Ramanujan Journal 31 (2013), no. 3, 271-279. DOI: https://doi.org/10.1007/s11139-012-9462-y. Google Scholar

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.