The number of zeros of linear combinations of \(L\)-functions near the critical line (Q6566354)

In this paper under review, the authors study the zeros near the critical line of a linear combination of \(L\)-functions in the Selberg class. More precisely, let \(J \geq 2\) be an integer, \(b_1,\ldots, b_J\) be non-zero real numbers such that \(\sum^J_{j=1} b^2_j = 1\), and let define \[F(s)=\sum_{j=1}^Jb_jL_j(s), \]where \(L_1,\ldots, L_J\) are distincts functions in the Selberg class satisfying mainly the assumptions of the Euler product, functional equation, the Ramanujan hypothesis, the zero density hypothesis and the Selberg orthogonality conjecture. The latter assumptions are standard, and are expected to hold for all \(L\)-functions arising from automorphic representations on \(\mathrm{GL}(n)\). \N\NThe main result of the paper is that the number of zeros \(N_F(\sigma,T)\) of \(F(s)\) in the region \(\Re(s) \geq \sigma=\frac{1}{2}+\frac{1}{G(T)}\) and \(\Im (s)\in [T, 2T]\) with \(\log\log T \leq G(T) \leq (\log T)^{\nu}\) satisfies \[N_F(\sigma,T)=K_0 \frac{TG(T)}{\sqrt{\log G(T)}}+ O\left(\frac{TG(T)}{\left(\log G(T)\right)^{5/4}}\right), \]\Nwhere \(K_0\) is a certain positive constant that depends on \(J\) and the \(L_j\) and the exponent \(\nu\) satisfies \(\nu \asymp 1/J\) as \(J\) grows.