Explicit formulas for the pair correlation of zeros of functions in the Selberg class (Q2769473)

The authors obtain several interesting results concerning explicit formulas and the pair correlation of zeros of functions in the Selberg class \(\mathcal S\) class of Dirichlet series [see the comprehensive article of \textit{J. Kaczorowski} and \textit{A. Perelli} for the relevant definitions and properties of \(\mathcal S\), in: Number Theory in Progress, K. Györy et al. (eds.), Proc. Conf. in Honor of A. Schinzel, de Gruyter, 953--992 (1999; Zbl 0929.11028)]. By an explicit formula for \(F \in {\mathcal S}\), one means a relationship of sums over primes of the form \(\sum_n \Lambda_F(n)f(n)\) (\(\Lambda_F(n)\) is the analogue of the von Mangoldt function \(\Lambda(n)\) for \(F\), generated by \(-F'(s)/F(s)\)), to sums over the zeros \(\rho\) of \(F\) of the form \(\sum_\rho g(\rho)\), where \(f\) and \(g\) are related by the Fourier transform. Assuming the weak pair correlation conjecture (this is the analogue, for \(F \in {\mathcal S}\), of the classical pair correlation conjecture of H.~L. Montgomery for the Riemann zeta-function), they prove that (i) one has unique factorization in \(\mathcal S\), (ii) the Artin conjecture on the holomorphy of non-Abelian \(L\)-functions is true, (iii) the Langlands reciprocity conjecture for solvable extensions of \(\mathbb Q\) is true. Other results, among others, include an (unconditional) explicit expression for the pair correlation function, an explicit expression for \(\sum_{n\leq x}\Lambda_F(n)\Lambda_G(n)\) for \(F,G \in {\mathcal S}\), and an explicit formula for \(\sum_{|\text{Im}\, \rho|\leq T}x^\rho\) (the generalized Landau-Gonek formula) when \(F \in {\mathcal S}\).