The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation ...

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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), ...
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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} ...
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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 ...
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4answers
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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 ...
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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 ?
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$\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$

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) ...
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1answer
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 ...
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1answer
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 ...
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2answers
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 ...
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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 ...
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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} ...
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1answer
827 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 ...
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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 $\sigma=\Theta$? ...
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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- ...
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1answer
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 ...
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1answer
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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 $$ ...
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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} ...
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1answer
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} ...
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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 ...
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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 $\Re(s) \ne \frac12$: ...
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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 ...
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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 ...
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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 ...
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2answers
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
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2answers
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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 ...
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2answers
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
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1answer
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
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4answers
819 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.
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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 ...
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2answers
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 ...
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2answers
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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 ...
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1answer
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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 ...
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1answer
604 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 ...
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1answer
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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 ...
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4answers
615 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 ...
33
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1answer
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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 ...
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2answers
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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 ...
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1answer
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 ...
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0answers
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 ...
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0answers
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 ...
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2answers
793 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
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2answers
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 ...
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1answer
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
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1answer
604 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')} ...
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4answers
957 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
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2answers
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 ...
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5answers
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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 ...
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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
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1answer
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.
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1answer
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 ...