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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 n^{2p}, 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 N_{t} (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).

title = "A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions",

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).",

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).