An asymptotic approximation for the Riemann zeta function revisited
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 $\omega$.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.07863
- arXiv:
- arXiv:2203.07863
- Bibcode:
- 2022arXiv220307863P
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- 11M06;
- 33B20;
- 34E05;
- 41A60
- E-Print:
- 8 pages, 0 figures