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

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Incomplete gamma Function
Riemann zeta function
Asymptotic Approximation
Summation
Directly proportional
Integer
Line
Term
Generalization

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A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions. / Paris, Richard B.

In: Applied Mathematical Sciences, Vol. 3, No. 60, 2009, p. 2973-2984.

Research output: Contribution to journalArticle

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AB - 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).

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