# Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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### How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?

If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$. Are ...
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### How fast does the proportion guaranteed by dirichlet converge?

I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
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### Why are the singular values equal to the first partial derivatives.

I am studying computer science so please go easy on me. I am also too bad at math to extract the mathematical essence that is needed to answer this question so I'm just gonna explain the whole setup. ...
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### Alternating Dirichlet series involving the Möbius function.

It is well known that: $$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$ with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function. Numerical ...
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### What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?

The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by $$A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p$$ is this serie calculated ...
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### $\sum_{n=1}^{+\infty}\frac{\Lambda\left(n\right)\varphi\left(n\right)}{n^{s}}$ and Riemann Zeta function
Is it possible to write $\sum_{n=1}^{+\infty}\frac{\Lambda\left(n\right)\varphi\left(n\right)}{n^{s}}$, where $\Lambda(n)$ is the Von Mangoldt function and $\varphi(n)$ is the Euler totient function, ...