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Dirichlet Series: Overview, Notation

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The Dirichlet series has the form [1]:
dirichlet series

Where:

The notation s is a long standing tradition (going back to Dirichlet) and is written as s = σ + it, where s is the real part and t is the imaginary part. When σ > 1, the series converges absolutely and serves as a generating function for f(n).

There are hundreds of thousands of infinite and finite Dirichlet series of all kinds [2]. Two of the most well known are the Riemann zeta function, defined as the Dirichlet series associated with the constant function 1, and Dirichlet L-functions [3].


Use of the Dirichlet Series

The series is often seen as a component for many generating functions for arithmetic functions. The method of generating functions is a powerful way of dealing with arithmetic functions. One type of generating functions is a type of power series:
f(0) + f(1)x + f(2)x2…,.
The other type is the Dirichlet series:
dirichlet generating function 2

The basic idea is that you relate the function f to the series F [3].

The Dirichlet series is a major component in proofs of the prime number theorem. For example, Hadamard and de la Vallee Poussin developed the first proof of the theorem by using the series to investigate the behavior of partial sums of arithmetic functions; Most proofs of the theorem relate the partial sums


to a complex integral involving the series

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References

[1] McCarthy, J. (2018). Dirichlet Series, Retrieved December 2, 2020 from: https://www.math.wustl.edu/~mccarthy/amaster-ds.pdf
[2] Gould, H. & Shonhiwa, T. A Catalog of Interesting Dirichlet Series. Retrieved May 4, 2021 from: https://projecteuclid.org/journalArticle/Download?urlid=10.35834%2Fmjms%2F1316032830
[3] Borcherds, R. (2021). Theory of Numbers: Dirichlet Series. https://youtu.be/dTQw2zt2YVw
[4] Hildebrand, A. (2005). Introduction to Analytic Number Theory. Math 531 Lecture Notes, Fall. Retrieved May 4, 2021 from: https://faculty.math.illinois.edu/~hildebr/ant/main4.pdf

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Stephanie Glen. "Dirichlet Series: Overview, Notation" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/dirichlet-series/
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