All Questions
Tagged with cv.complex-variables riemann-zeta-function
134 questions
5
votes
1
answer
766
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 ...
- 105k
-1
votes
1
answer
250
views
Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
- 105k
9
votes
0
answers
414
views
From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
-4
votes
1
answer
213
views
Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct? [closed]
The article can be freely accessed here. The proof is only five pages. I am quite in doubt.
A new version (2021) of that paper can be found here.
- 263
1
vote
0
answers
213
views
Convergence of zeta Euler product with additional term
Let's consider the following Euler product ($s=\sigma+it)$:
$$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$
So for $\sigma>1$, it is clear the product converges and we have:
$$...
- 1,199
2
votes
1
answer
561
views
On infinite sum containing logarithmic derivative of Zeta function and Möbius function:
Consider the following function:
$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$
Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...
- 149
35
votes
7
answers
6k
views
Heuristic argument for the Riemann Hypothesis
Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of ...
CommunityBot
- 1
3
votes
0
answers
128
views
Laplace transform of power of zeta function
Let $s$ is the complex variable. I would like to figure out the region of absolutely convergency of the following integral
$$
e^{\frac{is}{2}}\int\limits_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\...
- 11.1k
9
votes
1
answer
853
views
Moments of the Riemann zeta function
Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
- 105k
9
votes
1
answer
741
views
Three questions about three functions similar to $\sin,\cos$
In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
- 3,054
6
votes
1
answer
900
views
What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?
Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
- 114k
4
votes
1
answer
333
views
Double sum over zeros of Riemann zeta-function
In a paper by Saffari and Vaughan there appears a complicated-looking double sum
$$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
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12
votes
2
answers
1k
views
Has there been further work on Bender-Brody-Müller approach to RH?
Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
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23
votes
1
answer
3k
views
More mysteries about the zeros of the Riemann zeta function
Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.
Update on 1/5/2020: I added the section "more interesting ...
CommunityBot
- 1
-1
votes
1
answer
1k
views
Non trivial zeros of Riemann zeta function [closed]
Question Define $f(z)=(s-1)\zeta(s)$ where $s=\frac{1}{1+z^2}$ and $\zeta$ denotes the Riemann zeta function. Prove that if $\rho$ denotes the non trivial zeros of $\zeta(s)$ then, $$\sum_{|\alpha|&...
CommunityBot
- 1
0
votes
0
answers
151
views
Abscissa of convergence of transformed Dirichlet series
Let
$$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$
where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
- 3,098
2
votes
1
answer
672
views
Analytic continuation and convergence of a Riemann zeta related function
The functions in question are
$$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...
- 3,098
1
vote
0
answers
381
views
Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$
I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
- 3,098
0
votes
1
answer
379
views
On some property of the zeros of $\zeta(s)$ in the complex plane
This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well ...
- 362
2
votes
0
answers
210
views
Binomial transform of Dirchlet series (2)
Referring to this MO question, i managed to do the following :
We denote by $J(k+1,z)$ the sum :
$$J(n+1,z)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$
and by $S(k+1,z)$ the sum :...
- 181
4
votes
1
answer
925
views
On a possible equivalent of Riemann hypothesis
I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following :
The ...
- 63.9k
18
votes
2
answers
5k
views
How did Riemann calculate the first few non-trivial zeros of the zeta-function?
Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)...
17
votes
2
answers
2k
views
Algebraic independence of shifts of the Riemann zeta function
Let $\zeta(s)$ denote the Riemann zeta function. Is the set $\{
\zeta(s-j)\, \colon\, j\in\mathbb{Z}\}$, or even $\{\zeta(s-z)\,
\colon\, z\in\mathbb{C}\}$, algebraically independent over
$\mathbb{C}$?...
- 118k
0
votes
1
answer
607
views
On Soundararajan's explicit formula
I'm reading Soundararajan's https://arxiv.org/pdf/0705.0723.pdf, and on page 5, one has
$$\sum_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(...
- 1,019
5
votes
0
answers
161
views
On the asymptotics of some sum involving the Mertens function
Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...
- 1,019
3
votes
1
answer
528
views
Does the Riemann Xi function possess the universality property?
Here is the question.
Does the Riemann Xi function possess the universality property, or something similar to Voronin's universality property?
Here is why the answer to this question is important. ...
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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 ...
- 11.1k
4
votes
0
answers
126
views
What is the closed form of this integral?
Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, ...
- 41
9
votes
1
answer
939
views
A question on the Riemann zeta function
Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer,...
4
votes
4
answers
514
views
Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$?
let $P(x)\in{\mathbb Q}[x]{}$ be a rational polynomial with $P(1) >1$ and $\zeta $ be the Riemann zeta function , I want to know if there exist a rational polynomial such that $P(\zeta(s))=\zeta(P(...
- 28.2k
0
votes
0
answers
108
views
A line integral involving $\zeta(s)$
Let $x>1$ and $\zeta$ denote the Riemann zeta function. Is the equality
$$\int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\zeta(s)x^{s}}{s(s+1)(s+2)\cdots (s+k)} \mathrm{d}s = 2\pi i$$ possible ...
- 11
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
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
- 105k
3
votes
2
answers
1k
views
On the Dirichlet series for $1/\zeta(s)$ for real $s$ and the zeros of zeta
For $\Re(s)>1$, it is well known that
$$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$$ where $\mu$ denotes the Mobius function and $\zeta$ is the Riemann zeta function. I have heard ...
- 356
1
vote
0
answers
105
views
sum involving Riemann zeta function
In my work arose the following series:
$$g(s) = \sum_{n = 0}^\infty \frac{\log(\zeta(n+2))}{n+2}s^{n}.$$
It has radius of convergence $2$ and converges for $|s| \leq 2$ except at $s = 2$. I'm ...
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1
vote
0
answers
131
views
Looking for a citation for a result of Littlewood
I see that it is proven by Hardy in 1914 that there are an infinite number of zeros on the critical line. I also see that the Hardy and Littlewood conjectures appear in some papers they wrote ...
2
votes
0
answers
180
views
Multiple zeta values related to fractional calculus and an Appell polynomial sequence
There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
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-1
votes
1
answer
512
views
Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s>1/2$? [closed]
Consider the prime zeta function, defined for $\Re(s)>1$, by the infinite series
$$\sum_{p} p^{-s} = \sum_{m=1}^{\infty}\frac{\mu(m)}{m}\log \zeta(ms)$$ where $p$ denotes a prime, $\mu$ the Mobius ...
- 2,060
8
votes
2
answers
2k
views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
- 11.1k
1
vote
0
answers
62
views
Example of an integer $n_0$ such that $1+\sum_{k=2}^{n_0} \zeta(k)^s=0$ has repeated roots
After I was studying the exercise Problem 4.20 from [1] I was inspired to ask about next problem, where $\zeta(k)$ denotes, for integers $k>1$, particular values of the Riemann zeta function. And $...
2
votes
1
answer
2k
views
Books on complex analysis for self learning that includes the Riemann zeta function?
I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:
Analytic number theory : the connection between complex analysis and
...
- 91.8k
6
votes
2
answers
784
views
On some analytic property of the Riemann zeta function
Denote by $\zeta$ the Riemann zeta function. For $\Re(s)=\sigma>0$, it is well known that
$$\sum_{n\leq x} n^{-s} = \zeta(s) + \frac{x^{1-s}}{1-s}+ O(x^{-\sigma}).$$
But do there exist infinitely ...
- 12.8k
0
votes
0
answers
126
views
On an integral involving $\zeta(1/2 + i\tau)$
Denote by $\zeta$ the Riemann zeta function. Define $$F_{y}(x)= \int_{-\infty}^{\infty} \frac{x^{iu}\zeta(1/2 + it + iu)}{u^2 + y^2} \mathrm{d}u.$$
For some fixed real number $t$, is there any $y&...
CommunityBot
- 1
6
votes
1
answer
383
views
Tight error terms for partial sums $\sum_{n\leq x} 1/n^s$
(a) Let $s>1$, $x>0$ be real. Then it is not hard to see that
$$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$
basically because $x\mapsto 1/x^s$ is ...
- 105k
12
votes
4
answers
916
views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 2,821
1
vote
1
answer
756
views
An integral involving the argument of the Gamma function and the Riemann Hypothesis
Evaluate $$I=\int_{0}^{\infty} \frac{t\arg
\Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
Note that $I$ converges ...
- 105
-1
votes
1
answer
243
views
On a certain representation of the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
- 105k
-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 $\zeta$ ...
CommunityBot
- 1
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}...
- 105k
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 ...
- 13.4k
-3
votes
1
answer
245
views
An interesting phenomenon of the analytic continuation of Riemann zeta function [closed]
It is known that
$$\Gamma (s) \zeta (s)=\int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$
this function is valid only for $\Re{s}>1$.
However, if we ignore this restriction, and integrate by using
$$\frac{...
- 105k