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?



Explicit version of the Burgess theorem
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 $\log^{1/4}(q)$.


