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

Richard B. Paris

Research output: Contribution to journal › Article

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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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Paris, Richard B. / A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions. In: Applied Mathematical Sciences. 2009 ; Vol. 3, No. 60. pp. 2973-2984.
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Paris, RB 2009, 'A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions', Applied Mathematical Sciences, vol. 3, no. 60, pp. 2973-2984.

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 journal › Article

TY - JOUR

T1 - A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions

AU - Paris, Richard B.

PY - 2009

Y1 - 2009

N2 - 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).

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).

M3 - Article

VL - 3

SP - 2973

EP - 2984

JO - Applied Mathematical Sciences

JF - Applied Mathematical Sciences

SN - 1314-7552

IS - 60

ER -

Paris RB. A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions. Applied Mathematical Sciences. 2009;3(60):2973-2984.

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