Abstract
As a continuation of Chaps. 4 and 5, we study here properties of Hurwitz zeta functions and Dirichlet L-functions such as their analytic continuation and functional equation, and calculate their special values at negative integers. There are various proofs for the functional equation; here we explain the method using a contour integral. Although there would be a viewpoint that it would be too much to introduce a contour integral, it is interesting for its own sake and useful too, so we venture to derive the functional equation from a contour integral by a method to cut out the path of the integral.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Edmund Taylor Whittaker (born on October 24, 1873 in Southport, England—died on March 24, 1956 in Edinburgh, Scotland).
- 2.
George Neville Watson (born on January 31, 1886 in Devon, England—died on February 2, 1965 in Warwickshire, England).
References
Hurwitz, A.: Einige Eigenschaften der Dirichlet’schen Funktionen \(F(s) =\sum \left (\frac{D} {n} \right ) \cdot \frac{1} {n^{s}}\), die bei der Bestimmung der Klassenanzahlen binärer quadratischer Formen auftreten. Zeitschrift für Math. und Physik 27, 86–101 (1882). (Mathematische Werke I, 72–88)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Japan
About this chapter
Cite this chapter
Ibukiyama, T., Kaneko, M. (2014). Special Values and Complex Integral Representation of L-Functions. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_9
Download citation
DOI: https://doi.org/10.1007/978-4-431-54919-2_9
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54918-5
Online ISBN: 978-4-431-54919-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)