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Uniform asymptotic approximations for incomplete Riemann zeta functions. (English) Zbl 1091.33016
An incomplete Riemann zeta-function ${\cal Z}_1(\alpha,x)$ is examined, along with a complementary incomplete Riemann zeta-function ${\cal Z}_2(\alpha,x)$. These functions are defined by ${\cal Z}_1(\alpha,x)=\{(1-2^{1-\alpha})\Gamma(\alpha)\}^{-1}\int^x_0 t^{\alpha-1}(e' +1)^{-1}\,dt$ and ${\cal Z}_2(\alpha,x)=\zeta(\alpha)-{\cal Z}_1(\alpha,x)$, where $\zeta(\alpha)$ is the classical Riemann zeta function. ${\cal Z}_1(\alpha,x)$ has the property that $\lim_{x\to\infty}{\cal Z}_1(\alpha,x)=\zeta(\alpha)$ for $\text{Re }\alpha>0$ and $\alpha\ne 1$. The asymptotic behaviour of ${\cal Z}_1(\alpha,x)$ and ${\cal Z}_2(\alpha,x)$ is studied for the case $\text{Re }\alpha=\sigma>0$ fixed and $\text{Im }\alpha=\tau\to\infty$, and using Liouville-Green (WKBJ) analysis, asymptotic approximations are obtained, complete with explicit error bounds, which are uniformly valid for $0\le x <\infty$.

MSC:
33E20Functions defined by series and integrals
11M06$\zeta (s)$ and $L(s, \chi)$
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
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References:
[1] Berry, M. V.: The Riemann -- Siegel expansion for the zeta function: high orders and remainders. Proc. roy. Soc. London ser. A 450, 439-462 (1995) · Zbl 0842.11030
[2] Berry, M. V.; Keating, J. P.: A new asymptotic representation for $\zeta $(12+it) and quantum spectral determinants. Proc. roy. Soc. London ser. A 437, 151-173 (1992) · Zbl 0776.11048
[3] Berry, M. V.; Keating, J. P.: The Riemann zeros and eigenvalue asymptotics. SIAM rev. 41, 236-266 (1999) · Zbl 0928.11036
[4] Dingle, B.: Asymptotic expansions and converging factors. III. gamma, psi and polygamma functions, and Fermi-Dirac and Bose-Einstein integrals. Proc. roy. Soc. London ser. A 244, 484-490 (1958) · Zbl 0080.04302
[5] Edwards, H. M.: Riemann’s zeta function. (1974) · Zbl 0315.10035
[6] Kölbig, K. S.: Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. comp. 24, 679-696 (1970) · Zbl 0226.65011
[7] Kölbig, K. S.: Complex zeros of two incomplete Riemann zeta functions. Math. comp. 26, 551-565 (1972) · Zbl 0259.65052
[8] Y.L. Luke, The Special Functions and Their Approximations, vol. II, Mathematics in Science and Engineering, vol. 53, Academic Press, New York-London, 1969, xx+485pp. · Zbl 0193.01701
[9] F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. Reprinted by AK Peters, Wellesley, 1997. · Zbl 0303.41035
[10] Paris, R. B.; Cang, S.: An exponentially-smoothed gram-type formula for the Riemann zeta function. Methods appl. Anal. 4, 326-338 (1997) · Zbl 0957.11057
[11] Paris, R. B.; Cang, S.: An asymptotic representation for $\zeta $(12+it). Methods appl. Anal. 4, 449-470 (1997) · Zbl 0913.11033
[12] Temme, N. M.: The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. anal. 10, 757-766 (1979) · Zbl 0412.33001
[13] E.W. Weisstein, Lambert W-Function, From MathWorld --- A Wolfram Web Resource, http://mathworld.wolfram.com/LambertW-Function.html.