On an integral summable to $2\xi (s)/(s(s-1))$
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- by P. L. Walker PDF
- Math. Comp. 32 (1978), 1311-1316 Request permission
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 $2\xi (s)/s(s - 1) = {\pi ^{ - s/2}}\Gamma (1/2s)\zeta (s)$, where $s = 1/2 + it$, 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
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494 G. PÓLYA AND G. SZEGÖ, Aufgaben und Lehrsätze aus der Analysis, Springer-Verlag, Berlin, 1925. B. RIEMANN, "Über die Anzahl der Primzahlen unter einer gegebenen Grosse," Collected Works, Dover, New York, 1953.
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp. 32 (1978), 1311-1316
- MSC: Primary 10H05
- DOI: https://doi.org/10.1090/S0025-5718-1978-0491550-7
- MathSciNet review: 0491550