All Questions
8 questions
2
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0
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Reference request for literature on the following function--power counting zeta function
I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
- 388
1
vote
0
answers
102
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Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune
In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
0
votes
1
answer
274
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Relation between infinite product and regularized product
For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product
\begin{equation*}
\prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...
2
votes
0
answers
279
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computing a certain contour integral [closed]
I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
- 139
12
votes
4
answers
916
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non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 7,609
8
votes
3
answers
2k
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Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$
Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...
- 11.4k
106
votes
6
answers
19k
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Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
- 29k
74
votes
10
answers
18k
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Why does the Gamma-function complete the Riemann Zeta function?
Defining $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ yields $\xi(s) = \xi(1 - s)$ (where $\zeta$ is the Riemann Zeta function).
Is there any conceptual explanation - or ...
- 12.3k