On certain Dirichlet series built from the Fourier coefficients of modular functions (Q706076)

Let \(f\) be a nonconstant meromorphic modular function of rational weight \(k\) with multiplier system \(v\) of finite order on a subgroup \(\Gamma\) of finite index in \(SL_2(\mathbb Z)\). For \(z\in \mathbb H\), write \(f(z)=\sum_{n\geq h}a(n)q_M^n\), where \(q_M=e^{2\pi i z/M}\), \(M\) is the least positive integer with \( \left(\begin{smallmatrix} 1 & M \\ 0 & 1 \end{smallmatrix} \right)\in \Gamma \), and \(h\) is the order of \(f\) at infinity. Normalize \(f\) so that \(a(h)=1\). For a prime power \(p^n\), put \(A(p^n)=a(n+h)\), and then extend the function \(A\) multiplicatively to the whole \(\mathbb N\). Define a formal Dirichlet series by \(D_f(s)=\sum_{m\geq 1}A(m)m^{-s}\). The following theorem is proved: Let \(f\) be as above, and suppose in addition that \(a(n)\) is an integer for all \(n\), \(a(n+1)>0\), and \(f\) has no zeros or poles on \(\mathbb H\). Then \(D(s)\) has abscissa of convergence equal to 1, and has a meromorphic continuation to \(\sigma>0\), but cannot be meromorphically continued into a neighborhood of \(s_0=0\).