7
votes
0answers
181 views

Counting the Number of Integral Solutions to $x^2+dy^2 = n$

It is a well known result that the number of integer solutions $(x,y), x>0, y\ge 0$ to $x^2+y^2 = n$ is $\sum_{d|n}\chi(d)$, where $\chi$ is the nontrivial Dirichlet character modulo $4$ such that ...
1
vote
0answers
95 views

Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
1
vote
1answer
71 views

Multiplicative Dirichlet Series

How do I go about proving that where $f(k)$ is multiplicative that: $$\sum_{k\geq 1}{\frac{f({k})}{k^{s}}}=\prod_{p}\sum_{k\geq 0}{\frac{f(p^{k})}{p^{ks}}}$$ I first tried to use the fundamental ...
2
votes
1answer
128 views

Euler product of Dirichlet Series

For $n$ a positive integer, let $f(n)$ be the squarefree part of $n$. Find the Euler product for $\mathfrak D_{f}(s)$ where $\mathfrak D_{f}(s)$ is the Dirichlet Series of $f$.