On uniqueness in the extended Selberg class of Dirichlet series (Q2855894)

The extended Selberg class is the set of all Dirichlet series \(L(s)=\sum_{n=1}^\infty a(n)n^{-s}\) with \(a(1)=1\) satisfying the following axioms: {\parindent=8mm\begin{itemize}\item[(i)] for \(\text{Re}(s)>1\), \(L(s)\) is an absolutely convergent Dirichlet series; \item[(ii)] there is a nonnegative \(k\) such that \((s-1)^kL(s)\) is an entire function of finite order; \item[(iii)] \(L(s)\) satisfies a \textit{functional equation} of type NEWLINE\[NEWLINE\Lambda_L(s)=\omega\overline{\Lambda_L(1-\overline{s})},NEWLINE\]NEWLINE where \(\Lambda_L(s)=L(s)Q^s\prod_{j=1}^K\Gamma(\lambda_js+\mu_j)\) with positive real numbers \(Q\), \(\lambda_j\) and with complex numbers \(\mu_j\), \(\omega\) with \(\text{Re}(\mu_j)\geq 0\) and \(|\omega|=1\). NEWLINENEWLINE\end{itemize}} The authors show that two non-constant functions \(L_1\) and \(L_2\) from the extended Selberg class satisfying the same functional equation are identically equal, provided \(Z^+(L_1)\setminus G\subseteq Z^+(L_2)\) for a set \(G\) with NEWLINE\[NEWLINE\tau(G):=\limsup_{r\to\infty}\frac{|G\cap\{s:|s|<r\}|}{r}<\frac{\log 4}{\pi},NEWLINE\]NEWLINE where \(|A|\) counts the number of points in \(A\) with their multiplicity and \(Z^+(L)\) denotes the set of all nontrivial zeros of \(L\) counted with multiplicity. Moreover, it turns out that the inequality for \(\tau(G)\) is best possible.