An asymptotic approximation for the Riemann zeta function revisited

R B Paris

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arXiv · 2203.07863

An asymptotic approximation for the Riemann zeta function revisited

R B Paris

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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 $\omega$.

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R B Paris. 2022-03-15. An asymptotic approximation for the Riemann zeta function revisited. https://arxiv.org/abs/2203.07863

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