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An asymptotic representation for $\zeta \left(\frac{1}{2}+it\right)$. (English) Zbl 0913.11033

The authors give a representation for the Riemann zeta function $\zeta \left(s\right)$ as an absolutely convergent expansion involving incomplete gamma functions. The uniform asymptotics of the latter functions are then used to give an asymptotic representation for $\zeta \left(s\right)$ on the critical line. Roughly speaking, this expansion consists of the original Dirichlet series smoothed by a modified complementary error function together with a correction term, which has an expansion with coefficients that can be calculated to any required accuracy by simple recurrences. It is also shown that the expansion diverges like the familiar ‘factorial divided by a power’ dependence, which is also the case when such an expansion is based on the Riemann-Siegel formula as shown in the paper by M. V. Berry [Proc. R. Soc. Lond., Ser. A 450, 439-462 (1995; Zbl 0842.11030)].

The treatment here is an improvement of an earlier effort by the first author [R. B. Paris, Proc. R. Soc. Lond., Ser. A 446, 565-587 (1994; Zbl 0827.11051)]. Various numerical tests of the formula are also given, showing that, at least with regard to the numerical calculation of $\zeta \left(s\right)$ inside the critical strip, the method offers an interesting alternative to the Riemann-Siegel formula.

##### MSC:
 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33B20 Incomplete beta and gamma functions 34E05 Asymptotic expansions (ODE)