All Questions
76 questions
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
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
37
votes
2
answers
3k
views
$\zeta(0)$ and the cotangent function
In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^\...
- 105k
35
votes
7
answers
6k
views
Heuristic argument for the Riemann Hypothesis
Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of ...
- 3,699
29
votes
1
answer
2k
views
Riemann's attempts to prove RH
I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I ...
- 3,699
20
votes
1
answer
745
views
On the equation $\zeta(s) = F(s)+F(s+1)$
Define the function $F(s)$ as the Dirichlet series
$$
F(s) = \sum_{n=1}^\infty \frac{1}{(n+1)n^{s-1}},
$$
which converges for $\operatorname{Re}(s)>1$.
Has anyone seen/studied this function before? ...
- 2,549
18
votes
2
answers
5k
views
How did Riemann calculate the first few non-trivial zeros of the zeta-function?
Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)...
- 3,699
17
votes
2
answers
2k
views
Algebraic independence of shifts of the Riemann zeta function
Let $\zeta(s)$ denote the Riemann zeta function. Is the set $\{
\zeta(s-j)\, \colon\, j\in\mathbb{Z}\}$, or even $\{\zeta(s-z)\,
\colon\, z\in\mathbb{C}\}$, algebraically independent over
$\mathbb{C}$?...
- 50.8k
15
votes
5
answers
2k
views
Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
- 11.1k
14
votes
1
answer
1k
views
The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$
Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
- 161
13
votes
2
answers
725
views
Special values of $\zeta$ outside the real line and the critical strip
The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
- 17.6k
12
votes
2
answers
555
views
$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$
A bit of plotting suggests that $\zeta^{(k)}(s) < 0$ for all $s\in (0,1)$ and all integers $k\geq 0$. (Or, what is the same: $\zeta^{(k)}(s)$ has no zeroes on $(0,1)$.) Is there a brief, clean ...
- 20.2k
12
votes
4
answers
916
views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 7,609
12
votes
2
answers
1k
views
Has there been further work on Bender-Brody-Müller approach to RH?
Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
- 6,976
11
votes
0
answers
530
views
Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
- 2,480
9
votes
2
answers
1k
views
On the error term of the Riemann explicit formula
Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \...
- 1,019
9
votes
1
answer
853
views
Moments of the Riemann zeta function
Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
- 150
9
votes
1
answer
741
views
Three questions about three functions similar to $\sin,\cos$
In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
- 3,032
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,...
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
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
- 603
9
votes
0
answers
264
views
Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?
Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
- 2,480
8
votes
3
answers
1k
views
Objections to and arguments for the simplicity of all Riemann zeros
It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter ...
- 2,480
8
votes
2
answers
296
views
How to formalize the *loci of equal arg($\zeta(s)$)* ("isogones") in the near of a nontrivial root
(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...
- 5,342
8
votes
2
answers
2k
views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
- 11.1k
7
votes
2
answers
788
views
Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$
Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
- 1,019
7
votes
2
answers
719
views
On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$
I am interested in determining the behaviour of the the series/function
$$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$
near $s=0$. It is clear that $f(0)$ is undefined....
- 405
6
votes
1
answer
900
views
What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?
Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
- 149
6
votes
1
answer
4k
views
About the logarithmic derivative of the Riemann zeta function
Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
- 219
6
votes
1
answer
247
views
Convergence and meromorphic continuation of a Dirichlet series under RH
Consider a Dirichlet series $F(s) = \sum_{n\geq1} \frac{a_{n}}{n^{s}}$ such that $a_{n} = O(1)$. Suppose the Dirichlet series
$$\zeta(s)F(s)=\sum_{n\geq1} \frac{(a\star1)(n)}{n^{s}}$$
converges ...
- 611
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 ...
- 20.2k
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
5
votes
1
answer
426
views
Lindelöf hypotheses for derivatives of zeta
The Lindelöf hypothesis says that if we have:
$$\zeta(\sigma+iT)=\mathcal O(T^a)$$
Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
- 83
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
5
votes
1
answer
291
views
Asymptotics of the Liouville sum at the primes
Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
- 103
5
votes
0
answers
321
views
Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
- 20.2k
5
votes
0
answers
260
views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 63.9k
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
5
votes
0
answers
694
views
Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?
The inverse of the Weierstrass transform
expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
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
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 ...
- 774
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
$$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
- 465
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
4
votes
0
answers
821
views
One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
- 11
4
votes
0
answers
450
views
Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
- 41
3
votes
2
answers
813
views
Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes?
There are two proofs of
$$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$
which I'm aware of. I'll call the first one the Sieve proof and the second one ...
- 157
3
votes
2
answers
1k
views
On the Dirichlet series for $1/\zeta(s)$ for real $s$ and the zeros of zeta
For $\Re(s)>1$, it is well known that
$$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$$ where $\mu$ denotes the Mobius function and $\zeta$ is the Riemann zeta function. I have heard ...
- 39
3
votes
3
answers
273
views
Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$
Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
- 35
3
votes
1
answer
528
views
Does the Riemann Xi function possess the universality property?
Here is the question.
Does the Riemann Xi function possess the universality property, or something similar to Voronin's universality property?
Here is why the answer to this question is important. ...
3
votes
1
answer
571
views
On the series $\sum_{\rho}x^{\rho}\Gamma(\rho)/\Gamma(\rho+k),\,0<k<1$
Let $x>1$ be a real number. For a work I need to find an uniform estimation of the series the series $$\sum_{\rho}x^{\rho}\frac{\Gamma\left(\rho\right)}{\Gamma\left(\rho+k\right)}\tag{1}$$ where $\...
- 219
3
votes
1
answer
308
views
Zeros of the derivative of $\xi$
In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that
It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the ...
- 61