All Questions

The Dirichlet $\eta$-function is defined as: $$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \qquad \Re(s) > 0$$ and has the full analytical continuation: $$\eta(s) = \sum_{n=1}^N \frac{(-1)^{...

In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...

I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write $$\...

Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$ They are absolutely convergent in the half-plane $\sigma>...