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on Burgess-quality subconvexity exponent for Dirichlet $L$-functions ($C_{62b}$)

The statement

A Dirichlet character of level $q$ is an arithmetic function $\chi$ that is multiplicative, is defined by a character on $(\mathbb{Z}/q\mathbb{Z})^\ast$ on integers coprime to $q$ , and is $0$ on integers not coprime to $q$ . [Ked2007-def-character]

Such a character is called primitive if it is not induced from a smaller level. [Ked2007-def-primitive]

For a Dirichlet character $\chi$ , the associated Dirichlet $L$ -function is

L(s,\chi)\ :=\ \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s},

with Euler product for $\Re(s)>1$ . [Ked2007-def-L]

Define the pointwise subconvexity exponent

\mu_{\mathrm{Dir}}\ :=\ \inf\Bigl\{\theta\ge 0:\ L(1/2,\chi)\ll_{\varepsilon} q^{\theta+\varepsilon}\ \text{for all primitive }\chi\ (\bmod\ q)\Bigr\}.

This formulation is in the same conductor aspect used in modern Dirichlet subconvexity work. [PY2020-primitive-conductor] [PY2020-central-values-q-aspect]

We define

C_{62b}\ :=\ \mu_{\mathrm{Dir}},

the Burgess-quality subconvexity exponent for Dirichlet $L$ -values at the central point.

Burgess proved that, for all primitive characters $\chi$ modulo $q$ ,

L(1/2,\chi)\ \ll_{\varepsilon}\ q^{3/16+\varepsilon}.

[PY2020-burgess-3-16] In particular, $C_{62b}\le 3/16$ .

The best established range currently is

0\ \le\ C_{62b}\ \le\ \frac{3}{16}.

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.