An asymptotic approximation for the Riemann zeta function revisited

TL;DR

This paper revisits a representation of the Riemann zeta function using incomplete gamma functions, deriving an asymptotic expansion on the critical line that includes correction terms expressed in inverse powers of a complex parameter.

Contribution

The paper refines the asymptotic approximation of the Riemann zeta function by expressing correction terms in a more accessible inverse power series format.

Findings

Provides an asymptotic expansion for ζ(s) on the critical line as t→∞.

Expresses correction terms in inverse powers of ω with trigonometric coefficients.

Enhances the usability of the approximation for analytical and numerical purposes.

Abstract

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics of the incomplete gamma function produces an asymptotic-like expansion for $\zeta(s)$ on the critical line $s=1/2+it$ as $t\to+\infty$. The main term involves the original Dirichlet series smoothed by a complementary error function of appropriate argument together with a series of correction terms. It is the aim here to present these correction terms in a more user-friendly format by expressing then in inverse powers of $\omega$, where $\omega^2=\pi s/(2i)$, multiplied by coefficients involving trigonometric functions of argument $\omega$.