### Abstract

*s*) involving incomplete gamma functions with their second argument proportional to

*n*, where

^{2p}*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*).

Original language | English |
---|---|

Pages (from-to) | 2973-2984 |

Number of pages | 12 |

Journal | Applied Mathematical Sciences |

Volume | 3 |

Issue number | 60 |

Publication status | Published - 2009 |

### Fingerprint

### Cite this

*Applied Mathematical Sciences*,

*3*(60), 2973-2984.

}

*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.

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 -