Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function

Author: K. S. Kölbig
Journal: Math. Comp. 24 (1970), 679-696
MSC: Primary 65.25
MathSciNet review: 0273787
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function are calculated as functions of the upper limit $ \lambda $ of the definition integrals. It becomes apparent that not all, but only some, of the zero trajectories of the incomplete Riemann zeta function join a zero of the Riemann zeta function $ \zeta (s)$ for $ \lambda \to \infty $. The remaining trajectories at least in the region considered, approach the zero trajectories of the incomplete gamma function.

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] R. Putschbach, Untersuchungen über die unvollständige Riemannsche Zetafunktion, Inst. f. Praktische Mathematik (IPM), Techn. Hochschule Darmstadt, 1948?. (Unpublished.)
  • [3] Eugene Jahnke, Fritz Emde, and Friedrich Lösch, Tables of higher functions, 6th ed. Revised by Friedrich Lösch, McGraw-Hill Book Co., Inc., New York-Toronto-London; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1960. MR 0114317
  • [4] Friedrich Lösch and Fritz Schoblik, Die Fakultät (Gammafunktion) und verwandte Funktionen mit besonderer Berücksichtigung ihrer Anwendungen, B. G. Teubner Verlagsgesellschaft, Leipzig, 1951 (German). MR 0047832
  • [5] K. S. Kölbig, Über die Riemannsche Zetafunktion, Diplomarbeit, Inst. f. Praktische Mathematik (IPM), Techn. Hochschule Darmstadt, 1960. (Unpublished.)
  • [6] Francesco G. Tricomi, Funzioni ipergeometriche confluenti, Edizioni Cremonese, Roma, 1954 (Italian). MR 0076936
  • [7] F. G. Tricomi, Asymptotische Eigenschaften der unvollständigen Gammafunktion, Math. Z. 53 (1950), 136–148 (German). MR 0045253,
  • [8] K. S. Kölbig, DGAUSS--Double Precision Gaussian Integration, CERN 6000 Series Program Library D109, 1968. (Unpublished.)
  • [9] M. Gérard, NEWTON--Solution of $ n$ Simultaneous Non-Linear Equations in $ n$ Unknowns by Newton's Method, CERN 6000 Series Program Library C400, 1968. (Unpublished.)
  • [10] P. Franklin, "Calculation of the complex zeros of the function $ P(z)$ complementary to the incomplete gamma function," Ann. of Math. (2), v. 21, 1919, pp. 61-63.
  • [11] F. H. Gronwall, "Sur les zéros des fonctions $ P(z)$ et $ Q(z)$ associées à la fonction gamma," Ann. École Norm. Sup. (3), v. 33, 1916, pp. 381-393.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65.25

Retrieve articles in all journals with MSC: 65.25

Additional Information

Keywords: Incomplete Riemann zeta function, incomplete gamma function, complex zeros
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society