**3**

votes

**0**answers

239 views

### Alternating sums of the non-trivial zeros of $\zeta(s)$.

It is known that the infinite sum of the non-trivial zeros $\rho_n =\beta + \gamma_ni$ of $\zeta(z)$, when taken in pairs that are either conjugated or reflexive (they give the same outcome), ...

**1**

vote

**2**answers

320 views

### Does there exist a closed form for the factors of this infinite product ?

Assume $s,a \in \mathbb{C}, a \pm in \ne 0$.
The following infinite product nicely converges and can be expressed in a closed form:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{a+i n} ...

**2**

votes

**2**answers

337 views

### Riemann Z function, bounds on number of non-trivial zeros along horizontal lines, rather than vertical ones

Concerning the non-trivial zeros of the Riemann Zeta function, one can find quite a lot of literature on:
the rate of growth of the number of zeros along the vertical critical line,
the zero-free ...

**18**

votes

**4**answers

1k 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 "expectional zeros" of course first ...

**5**

votes

**2**answers

649 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 ?

**29**

votes

**3**answers

3k views

### $\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$

When I tested this in Mathematica, I had expected it to say it did not converge. However, I got this:
$$\prod_{n=1}^\infty n^{\mu(n)}=\frac{1}{4 \pi ^2}$$
Note: this is the reciprocal of (3) ...

**0**

votes

**1**answer

193 views

### Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far]
For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...

**5**

votes

**1**answer

464 views

### Is there a connection between the closed forms of these two infinite products?

Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$$\displaystyle ...

**3**

votes

**2**answers

603 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 ...

**15**

votes

**2**answers

2k views

### On the Universality of the Riemann zeta-function

Hi,
I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference.
First, recall Voronin's remarkable theorem ...

**4**

votes

**0**answers

406 views

### An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found
$$
P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\left[
{\rm li}(t^{\frac zn-s})
\right]^{x}_2
\tag{7}
...

**0**

votes

**1**answer

830 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 ...

**0**

votes

**0**answers

149 views

### What can be said about zeros of $\zeta(s)$ sharing the largest real part?

Specifically, if $\rho$ is such that $\zeta(\rho)=0$ and $\max_{\rho}\Re(\rho)= \Theta$, can anything interesting be said about the number/distribution of zeros on the vertical line $\sigma=\Theta$?
...

**5**

votes

**3**answers

579 views

### A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH?

Building on this question scaling the imaginary part of $\rho$s in infinite products, I like to conjecture that:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right) \left(1- ...

**2**

votes

**1**answer

219 views

### Upper bounds for $\zeta(s)$ on the critical line

In Graham and Kolesnik's "Van der Corput's Method of Exponential Sums" they mention the results of Watt (1989) who obtained $\zeta(1/2 + it) = O(t^{89/560 + \epsilon})$.
Is anyone aware of more ...

**15**

votes

**1**answer

501 views

### The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
...

**4**

votes

**4**answers

453 views

### What happens to $\zeta(s)$ when all its $\Im(\rho_n)$ are “scaled” linearly?

I found that the following infinite product with $\mu = a +n b i$ and a,b real, $s \in \mathbb{C}$:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu} \right) \left(1- \frac{s}{1-\mu} ...

**1**

vote

**1**answer

321 views

### $\zeta(2k+1)$ expressed in a product of two infinite products of non-trivial zeros.

Take the Hadamard product for $\zeta(s)$:
$$\displaystyle \zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} ...

**33**

votes

**6**answers

3k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**2**

votes

**0**answers

134 views

### The influence of $\chi(s)$ on complex zeros of $\frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$

I was exploring the formula:
$$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$
and found that for all $\Re(s) \ne \frac12$:
...

**6**

votes

**2**answers

1k views

### Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...

**1**

vote

**0**answers

249 views

### Mellin inverse of the Hadamard product rep. of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely :
$$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...

**8**

votes

**3**answers

576 views

### Sharpening of Lindelöf hypothesis

The Lindelöf hypothesis is:
$$
\forall \epsilon >0,\exists C_\epsilon >0,\forall t\ge 1,\quad
\vert\zeta(\frac12+it)\vert\le C_\epsilon t^\epsilon.\qquad \tag{LH}.
$$
It is a weaker statement ...

**3**

votes

**2**answers

340 views

### How to check numerical precision of my computation of Stieltjes-constants?

In a thread in MSE I proposed an older routine of mine for the efficient computation of coefficients; I use a very similar routine for the quick&dirty computation of the Stieltjes-constants.
...

**6**

votes

**2**answers

1k views

### Euler Mascheroni Constant, Curious?

I found this formula for the Euler-Mascheroni constant $\gamma$.
Just wondering whether such a formula already exists in literature?
Also, wanted to know whether there are formulas that ...

**0**

votes

**2**answers

408 views

### What are the fallacies that this RH inequality may fail at most finitely often?

According to "EQUIVALENCES TO THE RIEMANN HYPOTHESIS
p.4
Let $g(n)$ be the maximal order of a permutation of n objects
RH Equivalence 3.3. The Riemann Hypothesis is equivalent to
$\log{g(n)} < ...

**6**

votes

**1**answer

700 views

### If a non-trivial zero of the zeta function existed off the critical line, would infinitely many zeros exist with the same real part?

It is known that there exist infinitely many non-trivial zeros of the Riemann zeta function in the critical strip. Also, we know that infinitely many zeros are on the critical line - more than 1/3 ...

**8**

votes

**4**answers

820 views

### Axioms for Riemann $\zeta$ function

Are there any set of axioms that completely characterize the Riemann zeta function?
i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.

**1**

vote

**2**answers

711 views

### Trying to debunk a claim

Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$
Is there any already existing ...

**2**

votes

**2**answers

428 views

### explicit large gap for consecutive zeros of the Riemann zeta function

In Theorem 9.12, Titchmarsh (The Theory of the Riemann Zeta Function) proved that
For every large positive T, $\zeta(s)$ has a zero $\beta+i\gamma$ satisfying
$$
|\gamma-T|<\frac{A}{\log\log\log ...

**4**

votes

**2**answers

240 views

### Formula in common: How to search for same/similar equations in other knowledge domains?

Hi people
In a recent presentation by Sedgewick, he recounts in 1977 Flajolet noticed that they had a formula in common, both in different domains (see slide 4 in ...

**8**

votes

**1**answer

1k views

### What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?

Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis?
I've heard Freeman Dyson say that ...

**2**

votes

**1**answer

605 views

### Generalization of Mertens' theorem

One classical Mertens' theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$
It is now very natural to ask, whether we have some good estimate to ...

**13**

votes

**1**answer

920 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 ...

**11**

votes

**4**answers

616 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 ...

**34**

votes

**1**answer

2k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**10**

votes

**2**answers

3k views

### Current Status of the Riemann Hypothesis [closed]

Does anyone know the current progress in showing the Riemann hypothesis? I was only able to find this paper of Conrey that says at least 40% of the zeros of the Riemann Zeta function lie on the ...

**1**

vote

**1**answer

180 views

### Linear (in)dependence of $\Im(\rho_n)$ and fundamental theorem of arithmetic

Hello,
If I'm not mistaken, globally speaking, Riemann's explicit formula establishes a duality between prime numbers and the non trivial zeroes of the Riemann zeta functions. The imaginary parts of ...

**1**

vote

**0**answers

444 views

### How ..did Connes get it (trace formula)

i have been reading or at least trying to understand how Connes get the density (approximate) of states
$ N(E)= \frac{E}{2\pi}log \frac{E}{2\pi}- \frac{E}{2\pi}+ \frac{7}{8}+ \frac{1}{\pi}arg ...

**3**

votes

**0**answers

271 views

### Definite integral of $\zeta(s)$ over the critical strip

Take the following definite integral:
$$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$
with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0<\sigma<1$ and $t,\sigma \in \mathbb{R}$.
The graph of ...

**2**

votes

**2**answers

795 views

### A note by N. A. Carella on zero-free regions [closed]

http://arxiv.org/abs/0908.4287
I could not find any reviews for it, but if true its a major claim, because it says that $\Re(\rho) < 21/40$ where $\rho$ is a zeta zero.
My question:
Are there ...

**3**

votes

**2**answers

417 views

### Are all zeros of ζ^{k}(s)±ζ^{k}(1−s) on the critical line (k=k-th derivative)?

The non-trivial zeros of $\zeta^{k}(s)$, with $k=k^{th}$ derivative, do not lie on a line and seem to be distributed randomly in the region $\sigma > \frac12$. However the non-real zeros in the ...

**1**

vote

**1**answer

477 views

### Zeros of the function $\zeta(s) \pm \zeta(\overline s)$

Building on this question: Zeros of $\zeta(s) \pm \zeta(1-s)$, I experimented further with:
$$\zeta(s) \pm \zeta(\overline s)$$
Assuming $s=\sigma + ti$, I observed that this function also has many ...

**9**

votes

**1**answer

605 views

### Montgomery's pair correlation function without RH?

In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ...

**3**

votes

**4**answers

958 views

### Is this sum of reciprocals of zeta zeros correct?

I am trying to find or get a numerical approximation of
$$ \sum_{\rho \text{ non-trivial zeros of } \zeta} \frac{1}{\rho} $$
In The Riemann Hypothesis: Arithmetic and Geometry Lagarias gives the ...

**6**

votes

**2**answers

551 views

### Are there known non-real zeros of derivatives of Riemann zeta with 0 < Re(s) < 1/2?

According to New zero free regions for the derivatives of the Riemann zeta function
assuming the Riemann Hypothesis, $\zeta^{(k)}(s)$ has
at most a finite number of non-real zeros with ...

**17**

votes

**5**answers

1k views

### Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...

**0**

votes

**0**answers

124 views

### Expressing a polynomal as products of shifted Riemann zeta functions and reciprocals.

Suppose I have a polynomial $f$ in $p,t,$ say $f=1 \pm p^{a_1}t^{b_1} \pm \cdots \pm p^{a_s}t^{b_s}$ with $a_i,b_i \in \mathbb{N}\cup0$ and $b_i$ is non-zero. Let $X:=${$ \ (1-p^it^j) | i,j \in ...

**5**

votes

**1**answer

439 views

### Ihara zeta function

Is there a natural connection between the Ihara zeta function of a graph,
and (for instance) the Riemann zeta function of certain varieties over finite fields ?
Thanks.

**-1**

votes

**1**answer

559 views

### Is there information about the $\rho$'s hidden in the zeros of $\Re(\chi(s))$ ?

Take the symmetrical form of the completed Zeta-function:
$\displaystyle \chi(s) \zeta(s) = \chi(1-s) \zeta(1-s)$
with
$\chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2})$.
For $s=\sigma + ti$, I ...