Mean values of derivatives of $L$-functions in function fields: IV
Mean values of derivatives of $L$-functions in function fields: IV
In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For $��\geq1$ an integer, we compute the mean value of the $��$-th derivative of quadratic Dirichlet $L$-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.
Mathematics - Number Theory, moments of \(L\)-functions, quadratic Dirichlet \(L\)-functions, derivatives of \(L\)-functions, Zeta and \(L\)-functions in characteristic \(p\), random matrix theory, 11M38, 11M06, 11G20, 11M50, 14G10, function fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Curves over finite and local fields, Relations with random matrices, \(\zeta (s)\) and \(L(s, \chi)\), FOS: Mathematics, Zeta functions and \(L\)-functions of function fields, Number Theory (math.NT)
Mathematics - Number Theory, moments of \(L\)-functions, quadratic Dirichlet \(L\)-functions, derivatives of \(L\)-functions, Zeta and \(L\)-functions in characteristic \(p\), random matrix theory, 11M38, 11M06, 11G20, 11M50, 14G10, function fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Curves over finite and local fields, Relations with random matrices, \(\zeta (s)\) and \(L(s, \chi)\), FOS: Mathematics, Zeta functions and \(L\)-functions of function fields, Number Theory (math.NT)
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