A note on certain infinite products (Q2366795)

For a Dirichlet character \(\chi\pmod M\) and a sequence of integers \((a(n))_{n=1}^ \infty\), \(a(n)=O(n^ c)\) for some positive \(c\), let \((\text{Im }z >0)\) \[ f_ \chi(z)= \exp(2\pi iaz) \prod_{h=0}^{M-1} \prod_{n=1}^ \infty (1-\zeta_ M^ h q(\lambda)^ n)^{\chi(h)a(n)}, \tag{1} \] where \(q(\lambda)= \exp(2\pi iz/\lambda)\), \(\lambda>0\) and \(a\) is a real number. Let \(\Phi\) be the associated Dirichlet series \(\Phi(s)= \sum_{n=1}^ \infty a(n)n^{-s}\). The main result of the paper states that if \(\Phi\) can be continued analytically as a non-zero meromorphic function with a finite number of poles on the complex plane and if there exists a real number \(k\) such that \(f_ \chi(-1/z)= (z/i)^ k f_ \chi(z)\), then \((\lambda/M)^ 2\) is an integer, \(a=k=0\), and \(f_ \chi\) is given by \[ f_ \chi(z)= \prod_{m\mid(\lambda/M)^ 2} \psi_ \chi(mz)^{c(m)}, \] where \(\psi_ \chi(z)= \prod_{h=0}^{M-1} \prod_{n=1}^ \infty (1- \zeta_ M^ h q(\lambda)^ n)^{\chi(h)\chi(n)}\), and for \(m\mid(\lambda/M)^ 2\) the coefficients \(c(m)\) are integers satisfying the condition \(c(m)=\chi (-1)c(\lambda/M)^ 2/m)\). Conversely, let \((\lambda/M)^ 2\) be an integer and let \(c(m)\), for integers \(m\) dividing \((\lambda/M)^ 2\), be arbitrary integers such that \(c(m)= \chi(-1)c(\lambda/M)^ 2 /m\) for any divisor \(m\) of \((\lambda/M)^ 2\). Then \(f_ \chi(z)\) defined by (1) satisfies \(f_ \chi(-1/z)=f_ \chi(z)\).