Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
  • [2] A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
  • [3] G. PÓLYA AND G. SZEGÖ, Aufgaben und Lehrsätze aus der Analysis, Springer-Verlag, Berlin, 1925.
  • [4] B. RIEMANN, "Über die Anzahl der Primzahlen unter einer gegebenen Grosse," Collected Works, Dover, New York, 1953.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10H05

Retrieve articles in all journals with MSC: 10H05

Additional Information

Article copyright: © Copyright 1978 American Mathematical Society