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Zbl 0913.11033
Paris, R.B.; Cang, S.
An asymptotic representation for $\zeta(\frac{1}{2}+it)$. (English)
[J] Methods Appl. Anal. 4, No.4, 449-470 (1997). ISSN 1073-2772

The authors give a representation for the Riemann zeta function $\zeta(s)$ 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 $\zeta(s)$ 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 {\it M. V. Berry} [Proc. R. Soc. Lond., Ser. A 450, 439-462 (1995; Zbl 0842.11030)]. \par The treatment here is an improvement of an earlier effort by the first author [{\it 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 $\zeta(s)$ inside the critical strip, the method offers an interesting alternative to the Riemann-Siegel formula.
[P.Shiu (Loughborough)]

MSC 2000:: *11M06 Riemannian zeta-function and Dirichlet L-function
41A60 Asymptotic problems in approximation
33B20 Incomplete beta and gamma functions
34E05 Asymptotic expansions (ODE)

Keywords: Riemann zeta function; incomplete gamma functions; error function; asymptotic representation; Riemann-Siegel formula

Citations: Zbl 0842.11030; Zbl 0827.11051

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