Explicit version of the Burgess theorem

Question

Does there exist a totally explicit version of the Burgess theorem? Precisely, let $m$ be a positive integer, and let $\chi$ be a primitive character mod $m$. A special case (sufficient for my purposes) of the Burgess theorem asserts that

$\left| \sum_{a\le n\le a+x}\chi(n) \right|\ll_\varepsilon x^{1/2}m^{3/16+\varepsilon}$

I wonder if this was ever made totally explicit. In the case $m=p$ prime Iwaniec and Kowalski prove in their book the inequality with right-hand side $cx^{1/2}p^{3/16}(\log p)^{1/2}$ and claim (without going into details) that $c=30$ is nice.

I need, however, the case of composite modulus as well. Was anything like this ever done?

Answer 1

There are now at least two instances of such an explicit result.

1. Theorem 1.1 of Bordignon: https://arxiv.org/abs/2001.05114.
2. Theorem 1.1 (and Corollary 1.2) of Jain-Sharma, Khale, and Liu: https://arxiv.org/abs/2010.09530v2, or https://www.worldscientific.com/doi/10.1142/S1793042121500834.

Bordignon's result applies to convoluted Dirichlet characters; however, applying the result with $\psi$ identically $1$ (and $k=1$) gives the following

\begin{equation*} \left\rvert\sum_{M<n \leq M+N} \psi(n) \chi(n)\right\rvert \leq m d(q)^{3 / 2} N^{1 / 2} q^{3 / 16}(\log q \log \log q)^{\frac{1}{2}}, \end{equation*}

for $q \geq \max\{h^4, q_0\}$, and with $m,h,q_0$ given in a table on p. 3. Note that in Bordignon's result, the modulus $q$ can be taken as small as $10^5$.

Theorem 1.1 of Jain-Sharma, Khale, and Liu states that for $q \geq \exp(\exp(9.594))$ and $N \leq q^{5/8}$,

\begin{equation*} \left\rvert S(M,N)\right\rvert \leq \sqrt{N}q^{\frac{3}{16}} \cdot 9.07 \log^{\frac{1}{4}}(q) (2^{\omega(q)} d(q))^{\frac{3}{4}} \left(\frac{q}{\varphi(q)}\right)^{\frac{1}{2}}. \end{equation*}

Since $q/\varphi(q)$ is $O(\log\log q)$ this improves Theorem 1.1 of Bordignon by $\log^{1/4}(q)$.