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The integrated fourth moment of Dirichlet \(L\)-functions over rational function fields. (English) Zbl 07257233
Summary: We consider the hybrid fourth shifted moment of Dirichlet \(L\)-functions over rational function fields, where the moment average is taken over all odd primitive characters of modulus \(Q\in\mathbb{F}_q [t]\) and over the critical circle, which is the symmetry line of the corresponding functional equation. We obtain an asymptotic formula for this moment with the full main term for arbitrary modulus \(Q\), as \(\deg Q\to\infty\) and \(q\) is fixed. Moreover, in case of an irreducible modulus \(Q\) we get an exact formula for this hybrid moment.
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11T06 Polynomials over finite fields
11M50 Relations with random matrices
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11T55 Arithmetic theory of polynomial rings over finite fields
Full Text: DOI
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