A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions

Richard B. Paris

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    Abstract

    We derive an expansion for the Riemann zeta function ζ(s) involving incomplete gamma functions with their second argument proportional to n2p, where n is the summation index and p is a positive integer. The possibility is examined of reducing the number of terms below the value Nt (t/2π)1/2 in the finite main sum appearing in asymptotic approximations for ζ(s) on the critical line s = 1 2 + it as t → ∞. It is shown that the expansion corresponding to quadratic dependence on n (p = 1) is the best possible representation of this type for ζ(s).
    Original languageEnglish
    Pages (from-to)2973-2984
    Number of pages12
    JournalApplied Mathematical Sciences
    Volume3
    Issue number60
    Publication statusPublished - 2009

    Keywords

    • Asymptotics
    • Zeta function
    • Incomplete gamma functions

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