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Dirichlet Series

A Dirichlet series is a series of the form

suma(n)e^(-lambda(n)z),

where a(n) and z are complex and {lambda(n)} is a monotonic increasing sequence of real numbers. The numbers lambda(n) are called the exponents, and a(n) are called the coefficients. When lambda(n)=lnn, then e^(-lambda(n)z)=n^(-z), the series is a normal Dirichlet L-series. The Dirichlet series is a special case of the Laplace-Stieltjes transform.

See also

Dirichlet Generating Function, Dirichlet L-Series, Laplace-Stieltjes Transform, Modular Form, Modular Function, Zeta Function

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References

Apostol, T. M. "General Dirichlet Series and Bohr's Equivalence Theorem." Ch. 8 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 161-189, 1997.Bohr, H. "Zur Theorie der allgemeinen Dirichletschen Reihen." Math. Ann. 79, 136-156, 1919.

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Dirichlet Series

Cite this as:

Weisstein, Eric W. "Dirichlet Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirichletSeries.html

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