Tagged Questions
0
votes
0answers
40 views
Prove that $f(s)=εf(2-s)$
Let $f$ be an analytic function defined by
$$f(s)=N^{-s/2}(2π)^{s}Γ⁻¹(s)∑_{n=1}^{∞}(a_{n}/n^{s})(F_{n}(s-1)-εF_{n}(1-s))$$
where $$F_{n}(t)=Γ(t+1,2πn/√N).(√N/2πn))^{t+1},ε=±1$$ and ...
5
votes
1answer
91 views
Serre's proof that zeta function is meromorphic
I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as
\begin{align*}
...
1
vote
2answers
75 views
Estimating the integrated Tchebychev function and calculating its error
I would like to understand how to derive (2) from (1) below.
Problem:
If $\psi_1$ is the integrated Tchebychev function below
$$\psi_1(x)=\frac{1}{2\pi i} ...
2
votes
1answer
63 views
Derivative of the Selberg $\zeta$-function
I want to compute the derivative of the Selberg $\zeta$-function:
$$ \mathcal{Z}(s)=\prod_{\gamma \; \text{primitive}} \prod_{n=0}^\infty (1-e^{-l(\gamma)(n+s)}); \qquad \Re(s)>1.$$
Where ...
7
votes
2answers
567 views
Approximation of Products of Truncated Prime $\zeta$ Functions
The problem arose, while I was looking at products of power prime zeta functions
$$
P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks},
$$
with $k\in \mathbb{N}$ and $s=it$ with real $t$.
By ...
2
votes
1answer
152 views
Periodic Zeta Function Functional Equation
Recall that the periodic zeta function has the Dirichlet series
$$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$
This defines an analytic function for $\Re s>0$ and has a ...
4
votes
0answers
136 views
When functions, analytically continued, carry over certain properties
Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
5
votes
1answer
266 views
Square-free zeta function zeros
It is a well known fact that the geometric series
$$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)$$
...
