Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the reciprocity law for the twisted second moment of Dirichlet
  $L$-functions   We investigate the reciprocity law, studied by Conrey~\cite{Con07} and
Young~\cite{You11a}, for the second moment of Dirichlet L-functions twisted by
$\chi(a)$ modulo a prime $q$. We show that the error term in this reciprocity
law can be extended to a continuous function of $a/q$ with respect to the real
topology. Furthermore, we extend this reciprocity result, proving an exact
formula involving also shifted moments.
  We also give an expression for the twisted second moment involving the
coefficients of the continued fraction expansion of $a/q$, and, consequently,
we improve upon a classical result of Selberg on the second moment of Dirichlet
L-functions with two twists.
  Finally, we obtain a formula connecting the shifted second moment of the
Dirichlet $L$-functions with the Estermann function. In particular cases, this
result can be used to obtain some simple explicit exact formulae for the
moments.

25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th