All Questions
11 questions with no upvoted or accepted answers
14
votes
0
answers
831
views
Growth of residues of $1/\zeta(s)$: conjectures?
Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let
$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| =
\max_{|\Im \rho|\leq T} \frac{1}...
- 20.2k
6
votes
0
answers
338
views
Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following:
If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
- 1,805
5
votes
0
answers
320
views
Dirichlet series associated with polynomials
Yesterday, I was looking for references about meromorphic extensions of certain Dirichlet associated with polynomials. I was surprised to discover that much less than I previously thought was known.
...
- 1,805
5
votes
0
answers
97
views
Compensation by the residue of the zeta function
(Repost of a question from MSE, where it found no success)
Let $F$ be a global number field. Introduce a local quantity at every place
$$x_p = \frac{\zeta_p(1)}{\zeta_p(2)}$$
for instance. The ...
- 5,424
3
votes
0
answers
534
views
Serge Lang's proof of Brauer-Siegel theorem
I was reading through chapter 16 of Lang's Algebraic Number Theory book. The chapter is fully devoted to proving the Brauer-Siegel theorem: Let ${(k_n/ \mathbb{Q})}_n$ be a sequence of galois ...
- 577
2
votes
0
answers
79
views
Reference request for literature on the following function--power counting zeta function
I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
- 388
2
votes
0
answers
82
views
density of zeroes of Epstein zeta functions on vertical strips
There are many results (e.g. Davenport and Heilbronn) asserting that Epstein zeta functions in general have zeroes outside the line $\mathrm{Re\ } s = n/4$. Is there any result about the density of ...
- 91
1
vote
0
answers
482
views
Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 790
1
vote
0
answers
102
views
Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune
In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
1
vote
0
answers
202
views
Estimation of the $k$-th derivative zeta function
When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question:
Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ ...
- 1,257
0
votes
0
answers
101
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Relating the multiplicative Fourier transform and the derived characteristic polynomial
(Tuesday, Sept 5:) For a number field $Fˣ$ and a number ring $Oˣ$ it is common to define:
$Z(f,χ) = ʃ_{Fˣ} f(x) χ(x) dˣ x$
$g(ω,ψ) = ʃ_{Oˣ} ω(x) ψ(x) dˣ x$
where $dˣx$ is the multiplicative Haar ...
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