A simple note on the number-theoretic function \(\Omega (n)\) and Dirichlet \(L\)-functions (Q6044385)

For a Dirichlet character \(\chi\) and \(|z|<2\), set \[D_z(s,\chi):=\sum_{n=1}^\infty\frac{\chi(n)z^{\Omega(n)}}{n^s}.\] These series were introduced by \textit{A. Selberg} [J. Indian Math. Soc., New Ser. 18, 83--87 (1954; Zbl 0057.28502)]. In the paper under review, for non-principal \(\chi\), it is proved that for a positive integer \(N\), \(D_{1/N}(s,\chi)\) has analytic continuation to \(\Re(s)>\delta>1/2\) if and only if every zero of \(L(s,\chi)\) with real part greater than \(\delta\) has order divisible by \(N\). The proof is made upon an identity of the form \[L(s,\chi)=D_{1/N}(s,\chi)^NH(s),\] where \(H(s)\) is a non-vanishing analytic function in the half plane \(\Re(s)>1/2\).