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Distinct zeros of \(L\)-functions. (English) Zbl 0891.11044

Let \(L_1(s)\) and \(L_2(s)\) be ‘independent’ \(L\)-functions, satisfying certain basic hypotheses akin to the Selberg Conjectures. Let \(m_j (\rho)\) \((j=1,2)\) be the multiplicity of \(\rho\) as a nontrivial zero of \(L_j(s)\), and set \[ D(T,L_1,L_2) =\sum_{0<\text{Im} \rho\leq T} \max \bigl(m_1 (\rho)- m_2(\rho), 0\bigr). \] Then, subject to the above hypotheses, it is shown that \(D(T,L_1, L_2)\gg T\log T\).
In the case of Dirichlet \(L\)-functions, this was established by A. Fujii [Acta Arith. 28, 395-403 (1976; Zbl 0329.10028)]. However the present result includes \(GL_2\) \(L\)-functions, for example. The proof is motivated by the work of E. Bombieri and D. Hejhal [Duke Math. J. 80, 821-862 (1995; Zbl 0853.11074)].

MSC:

11M41 Other Dirichlet series and zeta functions
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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