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Questions tagged [dirichlet-series]

For questions on Dirichlet series.

In mathematics, a Dirichlet series is any series of the form $$ \sum_{n=1}^{\infty} \frac{a_n}{n^s}, $$ where $s$ and $a_n$ are complex numbers and $n = 1, 2, 3, \dots$ . It is a special case of general Dirichlet series.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann $\zeta$ function is a Dirichlet series with $a_n=1$, as also are the Dirichlet $L$-functions.

It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Johann Peter Gustav Lejeune Dirichlet.

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Calculating a Dirichlet Character

How would I calculate a function such as $$\sum_{n \leq x}r(n) = L(1, \chi) \cdot x + O(x^{1-\eta}),$$ where $r(n) = \sum_{d|n} \chi(d)$? The part I'm having difficulty calculating is the L-function part. This particular L-function is equal to…
The Traveling Salesman
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Two questions about Dirichlet series

Let $f(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}$ be a Dirichlet series with abscissa of absolute convergence $L$. Then, 1) Is there a formula for $L$ in terms of the coefficients $a_n$? 2) Must $f(s)$ have a singularity somewhere on the line…
Mustafa Said
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Absolutely convergent Dirichlet series with any given infinite number of roots

Suppose that $S = \{\rho_{j} \}$ is any sequence of complex numbers such that $0 < \text{Re}(\rho_{j}) < 1$, $|\rho_{j}| \to \infty$ as j goes to infinity. Then, can we construct a Dirichlet series $D(s) = \sum_{n \geq 1} a_{n}n^{-s}$ which…
Grown pains
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Multiplication of Dirichlet Series

here is the following theorem: "Given the two functions F(s) and G(s) represented by Dirichlet series with $L(s,f)=\sum_{n=1}^{\infty} \frac{f(n)}{n^s}$ for $\sigma>a$ and $L(s,g)=\sum_{n=1}^{\infty} \frac{g(n)}{n^s}$ for $\sigma>b$ then in the…
usere5225321
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How do we prove Dirichlet L-series converges when $\chi$ is non-trivial character and $s>0$?

Let $\chi$ be a non-trivial Dirichlet character modulo $a$. Also assume $a$ has a primitive root $r$. Prove that Dirichlet $L$ function $L(s,\chi)=\sum_{n=1}^\infty {\chi(n)\over n^{s}}$ converges when $s>0$. I think the key hint is the primitive…
user234902
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What are the periodic Dirichlet series?

My question is: What are the periodic Dirichlet series? Does the Riemann zeta function $ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ and the alternating zeta function $η(s)=\sum_{n=1}^\infty (-1)ⁿ⁻¹/n^{s}$ are periodic?
DER
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