All Questions
47 questions
106
votes
6
answers
19k
views
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
views
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
49
votes
3
answers
6k
views
The Hardy Z-function and failure of the Riemann hypothesis
David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
- 13.1k
40
votes
8
answers
12k
views
How does one motivate the analytic continuation of the Riemann zeta function?
I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated ...
- 3,806
40
votes
5
answers
8k
views
Is $\zeta(3)/\pi^3$ rational?
Apery proved in his paper from 1979 that $\zeta(3)$ is irrational, and we know that for all integers $n$,
$\zeta(2n)=\alpha \pi^{2n}$
for some $\alpha\in \mathbb{Q}$. Given these facts, it seems ...
- 7,013
29
votes
4
answers
5k
views
What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?
The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
- 2,200
29
votes
4
answers
5k
views
Good uses of Siegel zeros?
The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
- 291
20
votes
3
answers
2k
views
Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.
Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables ...
- 41.4k
18
votes
2
answers
1k
views
Can the Dedekind zeta function distinguish between real and imaginary quadratic number fields?
Suppose I am given a machine that gives me the coefficients $a_1$, $a_2$, $a_3$, ... of a Dirichlet series
$$\sum_1^{\infty} \frac{a_n}{n^s} $$
and assume that I know that this Dirichlet series is the ...
- 5,637
17
votes
3
answers
1k
views
PNT for general zeta functions, Applications of.
When I read it for the first time, I found the whole slog towards proving the Prime Number Theorem and the final success to be magnificent. So I am curious about more general results.
We talk of ...
- 7,442
15
votes
2
answers
1k
views
Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$
I am trying to find a formula for the following integral for non-negative integer $k$:
$$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$
My first thought was to use the formula $$\...
- 11.4k
15
votes
1
answer
1k
views
How do functional equations for zeta functions arise from the structure of a homology group?
I have read in various disparate sources that certain zeta functions satisfy functional equations as a consequence of some structure on some homology group. Here is an example of a quote in this ...
- 1,231
14
votes
1
answer
582
views
Growth of $\zeta_{\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]}(2)$
Let $K_n$ be the field $\mathbf Q[\cos(\frac{\pi}{2^{n+1}})]$ (the real subfield of the cyclotomic field $\mathbf Q[e^{\frac{i\pi}{2^{n+1}}}]$).
Is there anything known about the growth of the ...
- 1,980
14
votes
1
answer
1k
views
Is there an elegant algebraic proof of this formula for quadratic field discriminants?
Consider the Dirichlet series counting discriminants of real quadratic fields. Quadratic field discriminants are "basically" squarefree integers, so the associated Dirichlet series $\sum D^{-s}$ is "...
- 7,337
14
votes
2
answers
1k
views
What are zeta functions good for?
I know a couple of answers to the above question:
They can be used for point counting over finite fields/estimating the distribution of primes in characteristic 0.
There are various conjectures/...
14
votes
0
answers
831
views
Growth of residues of $1/\zeta(s)$: conjectures?
Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let
$$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
12
votes
4
answers
916
views
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
11
votes
2
answers
771
views
Tauberian theorem with better error term
This is a fairly vague question.
Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
- 1,362
11
votes
2
answers
2k
views
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
- 3,062
11
votes
1
answer
817
views
Lower bounds on zeta(s+it) for fixed s
This is most probably widely known and discussed here many times, so I am preliminay sorry.
Does Riemann conjecture imply some lower estimates on values, say $|\zeta(3/4+it)|$ for real $t$, when $|t|$...
- 109k
9
votes
1
answer
1k
views
Functional equation Dedekind zeta function
I'd like to know to what point is it possible to generalize
this method
for obtaining the functional equation for the Dedekind zeta function $\zeta_K(s)$ of a number field ?
Let $\mathfrak{C}$ be ...
- 3,403
8
votes
3
answers
2k
views
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
7
votes
2
answers
906
views
Positivity of the coefficients of Taylor series associated to the Riemann hypothesis
The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
- 7,057
6
votes
6
answers
2k
views
Sequences equidistributed modulo 1
Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results.
H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidistributed modulo 1.
I....
- 5,470
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
4
votes
1
answer
520
views
Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$
For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative.
I believe that the following statement is true:
$$\zeta_K\...
- 67
4
votes
2
answers
366
views
Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $
Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?
$$
\int_{k + 1/2}^{k + 3/2}
\frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
...
4
votes
1
answer
350
views
A principle around the Ramanujan's zeta function in short intervals
Here $s=\sigma+it$ denotes the complex variable. We denote the Ramanujan's zeta function
$$\varphi(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s}$$
for $\Re(s)>7$, where $\tau(n)$ is the Ramanujan tau ...
3
votes
1
answer
367
views
Weil Conjectures Analog for Multivariate Zeta Functions
We know that the Riemann zeta function can be generalized to multivariate zeta functions.
Is there a multivariate analog of the Weil conjectures?
- 13.9k
3
votes
0
answers
533
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
1
answer
461
views
How essential is the vanishing of the Dirichlet $L$-functions to Dirichlet's theorem on primes in arithmetic progressions?
I seem to recall that the prime number theorem (PNT) is equivalent to the fact that the Riemann zeta function $\zeta(s)$ is non-zero on all of $\text{Re}(s) = 1$ (see https://math.stackexchange.com/...
- 831
2
votes
1
answer
344
views
Connecting two pictures of the Zeta function
Lets consider two views of zeta functions of curves.
For the following, let $\mathbb{F}_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}_p}$ be the algebraic closure ...
user30211
2
votes
1
answer
263
views
The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$
Background: I'm facing the computation of the zeta regularization of the infinite product given by
$$\prod_{m=-\infty}^\infty (km+u)$$
for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. ...
- 562
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
1
answer
922
views
An integral representation of the Riemann zeta function
I am referring to the equality in equation $3.29$ (page 12) and $4.20$ (page 17) in this paper.
I am unable to recognize where this comes from or what is the general expression for values other than ...
- 1,893
1
vote
1
answer
181
views
For which number fields we know the nonexistence of Stark zeros?
Let $L$ be a number field and let $\zeta_L(s)$ be its associated Dedekind zeta function. It is known that $\zeta_L(s)$ has at most one zero in the region
$$1 - \frac1{4 \log d_L} \leq \sigma \leq 1, \...
- 13
1
vote
0
answers
482
views
Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 782
1
vote
0
answers
102
views
Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune
In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
1
vote
0
answers
201
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
1
answer
1k
views
Analytical continuation of the reciprocal of the Zeta function [closed]
Is the reciprocal of the Zeta function analytically continuable?
As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.
0
votes
1
answer
109
views
Expanding on a step in the calculation of ζ(f,χ,s) = Λ(s)
I'm trying to understand the derivation here:
Above, f is defined as a product of Schwartz-Bruhat functions which are their own Fourier transform.
I was hoping someone could spell out the second ...
user30211
0
votes
1
answer
239
views
Zeros of partial sums of the Ramanujan's zeta function
In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
0
votes
0
answers
101
views
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 ...
user30211
0
votes
1
answer
274
views
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}...
