All Questions
77 questions
5
votes
2
answers
850
views
Local phase statistics of the nontrivial Riemann zeros
(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...
- 9,501
-1
votes
1
answer
243
views
On a certain representation of the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
- 105
-2
votes
1
answer
270
views
A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$
By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that
$$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$.
where $\zeta$ ...
- 35
3
votes
3
answers
273
views
Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$
Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
- 35
0
votes
4
answers
715
views
On the real part of the Riemann zeta function inside the critical strip
Denote by $\zeta$ the Riemann zeta function. Does $\Re\zeta(s)$ ever vanish for $\frac{1}{2}<\Re(s)\leq 1$ ?
- 1,019
4
votes
2
answers
448
views
Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
- 395
52
votes
3
answers
6k
views
Is the Riemann zeta function surjective?
Is the Riemann zeta function surjective or does it miss one value?
- 2,375
0
votes
2
answers
339
views
Error term in França-LeClair approximation of zeta zeros
The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part (in increasing order) is asymptotically
$$
t_n \sim 2\pi\frac{n}{\log n}
$$
and has been ...
- 9,114
12
votes
2
answers
555
views
$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$
A bit of plotting suggests that $\zeta^{(k)}(s) < 0$ for all $s\in (0,1)$ and all integers $k\geq 0$. (Or, what is the same: $\zeta^{(k)}(s)$ has no zeroes on $(0,1)$.) Is there a brief, clean ...
- 20.2k
0
votes
1
answer
169
views
Analytic extension of the Hurwitz ζ function
For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ ...
- 1,241
3
votes
1
answer
571
views
On the series $\sum_{\rho}x^{\rho}\Gamma(\rho)/\Gamma(\rho+k),\,0<k<1$
Let $x>1$ be a real number. For a work I need to find an uniform estimation of the series the series $$\sum_{\rho}x^{\rho}\frac{\Gamma\left(\rho\right)}{\Gamma\left(\rho+k\right)}\tag{1}$$ where $\...
- 219
2
votes
1
answer
396
views
Is $|\zeta(e^{ni})|\leq \log(n)$ true for $n > 19$ and how do i can show it if it is?
I performed some computations in wolfram alpha looking at the behavior of the values of $|\zeta(e^{ni})|$ trying to predict a lower bound. I have got the following result:
For $n > 19 :|\zeta(e^{...
6
votes
1
answer
4k
views
About the logarithmic derivative of the Riemann zeta function
Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
- 219
8
votes
2
answers
296
views
How to formalize the *loci of equal arg($\zeta(s)$)* ("isogones") in the near of a nontrivial root
(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...
- 5,342
9
votes
0
answers
264
views
Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?
Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
- 2,480
15
votes
5
answers
2k
views
Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
- 11.1k
3
votes
1
answer
709
views
Residues and values of Riemann Zeta function at some points
I need the following computational results for proving something.
Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$,
i.e. $\gamma_0\sim 14.134...$.
1) what is ...
- 897
13
votes
2
answers
725
views
Special values of $\zeta$ outside the real line and the critical strip
The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
- 17.6k
5
votes
1
answer
765
views
Reference for Lindelöf Hypothesis implying finitely many zeros off critical line?
Can anyone give me a reference for the following theorem on the Riemann zeta function?
If the Lindelöf Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...
- 195
18
votes
2
answers
5k
views
How did Riemann calculate the first few non-trivial zeros of the zeta-function?
Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)...
- 3,699
9
votes
2
answers
2k
views
References on Taylor series expansion of Riemann xi function
I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
- 603
8
votes
2
answers
2k
views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
- 11.1k
2
votes
1
answer
2k
views
Can infinite polynomials be expressed as a product of its linear factors?
Background:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing $\...
- 487
16
votes
1
answer
2k
views
Certain functional equations for the Riemann Zeta function?
Referring to this question I asked on math.SE.
I am posting a more generalized question here, for answers and further inquiry.
For the Riemann zeta function, we know of the standard functional ...
- 731
12
votes
4
answers
916
views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 7,609
14
votes
1
answer
1k
views
On meromorphic continuation of zeta function(s) and special values at negative integers
Euler developped (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers.
In one ...
- 2,029
49
votes
4
answers
6k
views
If the Riemann Hypothesis fails, must it fail infinitely often?
That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...
- 17.6k