This is joint work with Min Lee and Nikos Diamantis. Let $f,g$ be modular forms of even weight $k$ for $SL(2,Z)$ with Fourier coefficients $a(n), b(n)$. We find a meromorphic continuation and two different spectral expansions for the double shifted multiple Dirrichlet series
$\sum_{n, h\geq 1} \frac{a(n+h) b(n) \sigma_{1-2v}(h)}{n^{s+k-1} h^{u}}$.
We then relate this to a smoothed average over $q$ close to $Q$ of
$$\sum_{\substack{\chi\bmod{q}, \\ \text{ primitive}}} L(1/2, f\times \overline{\chi}) L(1/2, g\times \chi).$$
and obtain a main term with a very sharp error term. Such a formula has been found for the special case of $q$ prime, in 2018, by Blomer, Fouvry, Kowalski, Michel,
, and Sawin. The presence of small prime divisors of $q$ make such a result very difficult for specific composite $q$, making an average over $q$ an attractive alternative.Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/11531