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on Polya-Vinogradov best constant (squarefree asymptotic) ($C_{72}$)

The statement

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}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.