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Dirichlet series
Let be an increasing sequence of positive real numbers tending to . A Dirichlet series with exponents is a series of the form
where and all the are complex numbers.
An ordinary Dirichlet series is one having for all . It is written
The best-known examples are the Riemann zeta function (in which is the constant ) and the more general Dirichlet L-series (in which the mapping is multiplicative and periodic).
When , the Dirichlet series is just a power series in the variable .
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 theory.
Let be a Dirichlet series.
1. If converges at , then converges uniformly in the region
where is any real number such that . (Such a region is known as a “Stoltz angle”.)
2. Therefore, if converges at , its sum defines a holomorphic function on the region , and moreover as within any Stoltz angle.
3. identically if and only if all the coefficients are zero.
So, if converges somewhere but not everywhere in , then the domain of its convergence is the region for some real number , which is called the abscissa of convergence of the Dirichlet series. The abscissa of convergence of the series , if it exists, is called the abscissa of absolute convergence of .
Now suppose that the coefficients are all real and nonnegative. If the series converges for , and the resulting function admits an analytic extension to a neighbourhood of , then the series converges in a neighbourhood of . Consequently, the domain of convergence of (unless it is the whole of ) is bounded by a singularity at a point on the real axis.
Finally, return to the general case of any complex numbers , but suppose , so is an ordinary Dirichlet series .
1. If the sequence is bounded, then converges absolutely in the region .
2. If the partial sums are bounded, then converges (not necessarily absolutely) in the region .
References
- 1 Jean-Pierre Serre. A Course in Arithmetic, chapter VI. Springer-Verlag, 1973. Zbl 0256.12001.
- 2 E. C. Titchmarsh. The Theory of Functions. Oxford Univ. Press, second edition, 1958. Zbl 0336.30001.
Mathematics Subject Classification
30B50 Dirichlet series and other series expansions, exponential series- Forums
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