On the reciprocity law for the twisted second moment of Dirichlet \(L\)-functions (Q2796080)

For \(a\) and \(q\) with \((a,q)=1\) and \(q>0\) we put NEWLINE\[NEWLINE M(a,q) =\frac{q^{1/2}}{\varphi(q)} \sum_{\chi\pmod{q}}^{\ast} |L(\tfrac12,\chi)|^2\chi(a), NEWLINE\]NEWLINE where the sum runs over primitive Dirichlet characters modulo \(q\). \textit{B. Conrey} [``The mean-square of Dirichlet \(L\)-functions'', Preprint, \url{arXiv:0708.2699}] obtained the approximate reciprocity relation for \(M(a,q)\) when \(a\) and \(q\) are primes with \(2\leq a<q\). Later, \textit{M. Young} [Forum Math. 23, 1323--1337, (2011; Zbl 1282.11109)] improved it in the following form: NEWLINE\[NEWLINE\begin{aligned} M(a,q)-M(-q,a) & =\frac{q^{1/2}}{a^{1/2}}\left(\log\frac{q}{a}+\gamma-\log(8\pi)\right)\\ & +\zeta(\tfrac12)^2\left(1-2\frac{q^{1/2}}{\varphi(q)}(1-q^{-1/2})+2\frac{a^{1/2}}{\varphi(a)}(1-a^{-1/2})\right)\\ & +E(a,q),\end{aligned} NEWLINE\]NEWLINE where \(a\) and \(q\) are primes with \(2\leq a<q\), \(\gamma\) is Euler's constant and \(E(a,q)\) is an error term. In fact, Young showed \(E(a,q)\ll aq^{-1+\varepsilon}+a^{-C}\) for any fixed \(\varepsilon\), \(C>0\).NEWLINENEWLINEIn this paper, the author shows that \(E(a,q)\) is a function of \(a/q\). That is, there is a continuous function \(E(x)\) such that it satisfies \(E(a/q)=E(a,q)\) and \(E(x)=O(x)\) as \(x\to 0^{+}\). Furthermore, the author shows that, roughly speaking, the discontinuity and unsmoothness of \(M(a,q)\) (as a function of \(a/q\)) can be described in terms of certain reciprocal shifted twisted moments of Dirichlet \(L\)-functions. The author mentions an application to the second moment of Dirichlet \(L\)-functions with two twists and a possible application to the fourth moment of Dirichlet \(L\)-functions at the central point.