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Mathematics of Computation

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On an integral summable to $ 2\xi (s)/(s(s-1))$

Author: P. L. Walker
Journal: Math. Comp. 32 (1978), 1311-1316
MSC: Primary 10H05
MathSciNet review: 0491550
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Abstract: Let $ \psi (x) = \Sigma _{n = 1}^\infty {e^{ - {n^2}\pi x}}$, and $ \chi (u) = {e^{u/2}}(1 + 2\psi ({e^{2u}}))$. The divergent integral $ 2\smallint _0^\infty \chi (u)\cos $ tu du is shown to be summable for certain complex values of t to the function $ 2\xi (s)/s(s - 1) = {\pi ^{ - s/2}}\Gamma (\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} s)\zeta (s)$, where $ s = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} + $ it, and $ \zeta (s)$ is the zeta-function of Riemann. The values of a resulting approximation to $ 2\xi (s)/s(s - 1)$ are computed and its zeros located.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1978 American Mathematical Society

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