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Joint extreme values of \(L\)-functions. (English) Zbl 1504.11093

In the theory of zeta functions, one natural problem is to find extreme values of the zeta function on a given vertical line lying in the right half of the critical strip. More precisely, It is well known that if \(L\) is a function in the Selberg class satisfying the Selberg orthonormality conjecture (SOC) then on every vertical line \(s = \sigma + it\) with \(\sigma \in (1/2, 1)\), the \(L\)-function takes large values of size \(\exp (c (\log t)^{1-\sigma}/\log \log t)\) inside a small neighborhood. In the paper under review, the authors prove a similar result by showing the existence of closely spaced extreme values on a given vertical segment, for any collection of \(L\)-functions \(L_1, L_2,\dots, L_k\) in the Selberg class which have polynomial Euler product and satisfy he Selberg orthonormality conjecture.

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11M41 Other Dirichlet series and zeta functions
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References:

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