Dirichlet series

Let (λn)n1 be an increasing sequence of positive real numbers tending to . A Dirichlet seriesMathworldPlanetmath with exponentsPlanetmathPlanetmath (λn) is a series of the form


where z and all the an are complex numbersMathworldPlanetmathPlanetmath.

An ordinary Dirichlet series is one having λn=logn for all n. It is written


The best-known examples are the Riemann zeta functionMathworldPlanetmath (in which an is the constant (http://planetmath.org/ConstantFunction) 1) and the more general Dirichlet L-series (in which the mapping nan is multiplicative and periodic).

When λn=n, the Dirichlet series is just a power seriesMathworldPlanetmath in the variable e-z.

The following are the basic convergence properties of Dirichlet series. There is nothing profound about their proofs, which can be found in [1] and in various other works on complex analysis and analytic number theoryMathworldPlanetmath.

Let f(z)=nane-λnz be a Dirichlet series.

  1. 1.

    If f converges at z=z0, then f converges uniformly in the region

    (z-z0)0  -αarg(z-z0)α

    where α is any real number such that 0<α<π/2. (Such a region is known as a “Stoltz angle”.)

  2. 2.

    Therefore, if f converges at z0, its sum defines a holomorphic functionMathworldPlanetmath on the region (z)>(z0), and moreover f(z)f(z0) as zz0 within any Stoltz angle.

  3. 3.

    f=0 identically if and only if all the coefficients an are zero.

So, if f converges somewhere but not everywhere in , then the domain of its convergence is the region (z)>ρ for some real number ρ, which is called the abscissa of convergence of the Dirichlet series. The abscissa of convergence of the series f(z)=n|an|e-λnz, if it exists, is called the abscissa of absolute convergence of f.

Now suppose that the coefficients an are all real and nonnegative. If the series f converges for (z)>ρ, and the resulting function admits an analytic extension (http://planetmath.org/AnalyticContinuation) to a neighbourhood of ρ, then the series f converges in a neighbourhood of ρ. Consequently, the domain of convergence of f (unless it is the whole of ) is boundedPlanetmathPlanetmathPlanetmath by a singularity at a point on the real axisMathworldPlanetmath.

Finally, return to the general case of any complex numbers (an), but suppose λn=logn, so f is an ordinary Dirichlet series annz.

  1. 1.

    If the sequenceMathworldPlanetmath (an) is bounded, then f converges absolutely in the region (z)>1.

  2. 2.

    If the partial sums n=klan are bounded, then f converges (not necessarily absolutely) in the region (z)>0.


  • 1 Jean-Pierre Serre. A Course in Arithmetic, chapter VI. Springer-Verlag, 1973. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0256.12001Zbl 0256.12001.
  • 2 E. C. Titchmarsh. The Theory of Functions. Oxford Univ. Press, second edition, 1958. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0336.30001Zbl 0336.30001.
Title Dirichlet series
Canonical name DirichletSeries
Date of creation 2013-03-22 13:59:22
Last modified on 2013-03-22 13:59:22
Owner bbukh (348)
Last modified by bbukh (348)
Numerical id 8
Author bbukh (348)
Entry type Definition
Classification msc 30B50
Related topic DirichletLFunction
Related topic RiemannZetaFunction
Related topic DirichletLSeries
Defines ordinary Dirichlet series
Defines Stoltz angle
Defines abscissa of convergence
Defines abscissa of absolute convergence