**13**

votes

**1**answer

877 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

585 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 ...

**25**

votes

**1**answer

994 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

177 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

431 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

266 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

744 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

410 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

458 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

565 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

859 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

536 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 ...

**15**

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

123 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

420 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

546 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 ...

**0**

votes

**3**answers

547 views

### Possible locations for non trivial zeroes lying off the critical line

It has been proven that:
1) if $s$ is a non trivial zero $\rho$ of $\zeta(s)$ then so is $1−s$.
2) $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
3) $ 0 < \Re(\rho) ...

**5**

votes

**2**answers

549 views

### Upper bounds on the difference of consecutive zeta zeros

There are many results on the spacing of the gaps between nontrivial zeros of the $\zeta$ function, from trivial (average value is $\frac{2\pi}{\log\gamma_n}$) to difficult (bounds on max and min ...

**13**

votes

**0**answers

782 views

### Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...

**9**

votes

**1**answer

1k views

### The Riemann's Zeta Function represented as a continued fraction and a question of convergence.

The Riemann's zeta function can be expressed as a continued fraction as follows
\begin{align*}
\zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty ...

**27**

votes

**4**answers

3k 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 ...

**19**

votes

**6**answers

3k views

### explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

**5**

votes

**1**answer

576 views

### The Correlation of the Mobius Function and Dirichlet Characters.

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
...

**3**

votes

**1**answer

545 views

### Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function.

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...

**2**

votes

**2**answers

326 views

### lower bound for $\Re\zeta(1+it)$

Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks

**2**

votes

**0**answers

318 views

### Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...

**15**

votes

**1**answer

746 views

### Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

http://math.stackexchange.com/questions/64566/riemanns-zeta-function-and-the-uniform-distribution-on-1-0
Stackexchange isn't getting really excited about this, so here it is.
The $n$th cumulant of ...