Tagged Questions
1
vote
0answers
37 views
What's the probability to find a value of $t<T$ where $|P_k(it)|<\epsilon$?
Given $P_k$, the truncated Prime $\zeta$ function, defined like
$$
P_k(it):=\sum_{n=1}^k p_n^{it},
$$
where $p_n$ is the $n$th prime. What's the probability to find a value or range of $t$ less than ...
0
votes
1answer
29 views
Is the Mean Value of $|P_k(it)|$ equal to $\sqrt{k}$?
Let $P_k$ be the truncated Prime $\zeta$ function, like
$$
P_k(it)=\sum_{n=1}^k p_n^{it},
$$
with $p_n$ being the $n$th prime. Numerics seem to indicate that the mean value of $|P_k|$ taken over all ...
2
votes
0answers
49 views
Gram's series for integral equation
The prime counting function $ \pi(x) $ satisfies the integral equation
$$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$
and it has the solution in terms of Gram's ...
2
votes
1answer
253 views
Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?
Motivation:
I was looking at the approximation of the truncated Prime $\zeta$ function
$$
P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right)
$$
(to be found here with or ...
5
votes
1answer
263 views
Square-free zeta function zeros
It is a well known fact that the geometric series
$$1+x+x^2+x^3+\ldots$$
has the following form
$$\frac{1}{1-x}$$
Another possible representation is
$$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$
...
