[Search][Subject Index][MathMap][Tour][Help!]

	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 11M: Zeta and L-functions: analytic theory

Introduction

History

Applications and related fields

Subfields

• 11M06: zeta(s) and L(s, chi)
• 11M20: Real zeros of L(s, chi); results on L(1, chi)
• 11M26: Nonreal zeros of zeta (s) and L(s, chi); Riemann and other hypotheses
• 11M35: Hurwitz and Lerch zeta functions
• 11M36: Selberg zeta functions and regularized determinants
• 11M38: Zeta and L-functions in characteristic p [new in 2000]
• 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
• 11M45: Tauberian theorems, See also 40E05
• 11M99: None of the above but in this section

Parent field: 11: Number Theory

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

This is one of the more commonly-studied areas of number theory. Among the many choices of text:

• Titchmarsh, E.: "Theory of the Riemann zeta-function", Clarendon, Oxford, 1951; second edition, Oxford Univ. Press, New York, 1986; MR 88c:11049
• Ivic, A.: "The Riemann zeta-function", Wiley, New York, 1985; MR 87d:11062
• Vaughan, R. C., "The Hardy-Littlewood method", Cambridge University Press, Cambridge-New York, 1981. 172 pp. ISBN 0-521-23439-5
• Karatsuba, Anatoly A., "Complex analysis in number theory", CRC Press, Boca Raton, FL, 1995. 187 pp. ISBN 0-8493-2866-7

See also the references for number theory in general.

Software and tables

Other web sites with this focus

• Dan Bump has a Riemann Hypothesis page.
• L-functions [Brian Conrey]

Selected topics at this site

• Euler and the birth of analytic number theory
• "Proof" of closed formulae for zeta(2n) (e.g. Sum (1/n^2) = pi^2/6 )
• Computing values of the Riemann Zeta function
• Stieltjes' expansion of zeta(s) near s=1
• Images of zeta along vertical lines
• Evaluating zeta and finding zeros
• Locations of zeros of zeta
• A sampling of recent work on the Riemann zeta-function and the Riemann hypothesis
• What's new with the Riemann Hypothesis?
• What is the Generalized Riemann Hypothesis?
• Estimating the number of primes less than N with zeros of the zeta function.
• Connection between zeros of zeta and the error term in the PNT
• Generalizations of the Riemann zeta function: Polygamma function, Hurwitz zeta, etc
• Adamchik's formula for the Hurwitz zeta function
• Dirichlet series related to zeta function and arithmetic functions
• Comments on Jacobi sums (Sum exp( - n(n+a)/2 ). )
• Computing psi(x)=Gamma'(x)/Gamma(x) at rational points.
• Artin's conjecture on zeta functions of number fields.
• Sums and products taken over all primes, involving Dirichlet characters
• Values of certain Dirichlet L-functions
• Inverting a discrete convolution
• (Tauberian) types of convergence of sequences