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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:
 33E20 Functions defined by series and integrals 11M06 $\zeta (s)$ and $L(s, \chi)$ 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
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##### References:
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