Tagged Questions
5
votes
1answer
105 views
Interesting phenomenon with the $\zeta(3)$ series
I noticed that if one takes certain partial sums of the series for $\zeta(3)$:
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$
an interesting phenomenon occurs ...
2
votes
1answer
149 views
Dirichlet character over Riemann zeta function
Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
3
votes
2answers
162 views
Approximate Riemann zeta function
Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.
In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.
My question is: Is there a Functional equation for ...
5
votes
0answers
169 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
5
votes
2answers
378 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
10
votes
2answers
310 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
