0
votes
1answer
58 views

Riemann Zeta circularity?

In this post I show: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Wolfram Alpha shows an alternate form for the primes: $$\frac{p_n{}^2}{p_n{}^2-1}=\frac{\left(\sum _{k=1}^{2^n} ...
2
votes
1answer
42 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
1
vote
0answers
63 views

Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
0
votes
1answer
153 views

what the RH equivalent for Riemann prime formula $\Pi(x)$?

Question follow the one answered already, zeros about Riemann Zeta function and some L-function Let's me try my best to make it clear on what I am asking. In his 1859 paper "On the Number of Primes ...
1
vote
0answers
60 views

How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: $$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
6
votes
1answer
97 views

The use of log in the Mean density of the nontrivial zeros of the Riemann zeta function (part 2)

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
1
vote
0answers
102 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
1
vote
1answer
80 views

Relation between $1-(n^{p-1}\mod p)$ and Riemann $\zeta$

Taking: $$\mathcal V_p=1-(n^{p-1}\mod p)$$ with $$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$ and Riemanns' well known functional equation, I get easily to this result: ...
0
votes
1answer
64 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from ...
0
votes
0answers
34 views

Is there a pair correlation function for $\log p$?

Recently I raised two independent questions: (1) Is there a pair correlation function for primes? (2) How $\log p$ distributed? Taking the answer of both let me raise a third question. Question: ...
2
votes
1answer
100 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
7
votes
1answer
217 views

Can exceptionally large primes be used to get information on the roots of $\zeta$?

Take a list $L$ of roots of the $\zeta$ function, like the ones provided by Andrew Odlyzko and plug it into the Prime Counting Function $\pi(x)$ given by $$ \pi(x) \approx \operatorname{R}(x^1) - ...
2
votes
0answers
249 views

Riemann $\zeta(s)$ non-trivial zeros on $Re(s)=1/2$ and the “spin” of the primes? [closed]

Can this conjecture be true? Let me intro before getting to the conjcture, but for those who like to go straight, see the bold paragraph at the end. Recently I was reading through the well known ...
0
votes
0answers
52 views

Transformation prime to Riemann zeta $p\rightarrow\zeta(s)$?

May you help for which condition the following transformation $p\rightarrow\zeta(s)$ holds $$\log \zeta(s)=\log p\left(\sum_{n=2}^\infty \frac{1}{\log n}\frac{1}{n^s}\right)$$
0
votes
4answers
104 views

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
1
vote
1answer
64 views

Is there a pair correlation function for primes?

Montgomery's pair correlation function for the non-trivial zeros of the Riemann $\zeta(s)$ function is defined via the term $$1- \left( \frac {\sin(\pi u)}{\pi u} \right)^2$$ Does anybody know if ...
-1
votes
3answers
165 views

Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
4
votes
1answer
301 views

The meaning of the Euler Formula for Zeta

Does anybody know about a "meaning" behind the Euler Formula, what does it really say about the primes? I know that it is in equation to the zeta function and also how it is derived, but cannot find ...
11
votes
2answers
559 views

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all?

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?
22
votes
3answers
2k views

Two Representations of the Prime Counting Function

The bounty for the best work out of Greg's answer, especially the "solving for $\pi^*(x;q,a)$ in terms of all $\Pi^*$ functions (tedious but possible)" part is over. Since Raymond's ...
1
vote
0answers
128 views

Approximation of distribution of $\pi_k(n)$ using $\zeta(s)$

Let $\pi_k(n) $ be the number of numbers with k prime factors (repetitions included) less than or equal to n. If we take the sums: $z_1(s) = \sum_{n= 1}^\infty \frac{1}{(p_{1,n})^s},~ z_2(s) = \sum ...
0
votes
0answers
50 views

Riemann Siegel formula modification?

$$Z(t)= \sum_{n=1}^{\lfloor\sqrt t/2\pi\rfloor }\frac{\cos(N(t)-t\log p_{n})}{\sqrt n}$$ here $N(t)$ is the smooth part of the zeros and $ p_{n} $ are the primes since $ p_{n} =n\log n $ then $ \log ...
3
votes
2answers
124 views

Why is the probability that a prime p is a factor of a number n equal to 1/p

I'm learning some number theory and I can't seem to understand why this is the case.
1
vote
1answer
150 views

Two Representations of $\log \zeta$

I was looking for representations of $\log \zeta$ and found these two: $ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
2
votes
1answer
213 views

is it possible to get the Riemann zeros

since we know that the number of Riemann zeros on the interval $ (0,E) $ is given by $ N(E) = \frac{1}{\pi}\operatorname{Arg}\xi(1/2+iE) $ is then possible to get the inverse function $ N(E)^{-1}$ ...
5
votes
1answer
255 views

Does this $\zeta(s)$ identity have a name?

I have generalized the product from this thread: Let $s=2n+1$ for $n\ge1$, $$\zeta (s)=\frac{\zeta (2 s)}{\zeta (2)} \prod _{n=1}^{\infty } \frac{\sum _{i=0}^{s-1}(-p_n){}^i}{(p_{n}-1)p_{n}^{s-2}}$$ ...
2
votes
1answer
250 views

Factor of the Euler Product at the Roots Of Zeta

The $\zeta$ function maybe written as Euler Product: $$ \zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}=\prod_p e_p(s). $$ Now let's substitute $s$ with $\rho_k$, the $k$th root of $\zeta$, and have a look at ...
7
votes
5answers
438 views

Rational Roots of Riemann's $\zeta$ Function

A look at the first few zeros $$14.134725,21.022040,25.010858,30.424876,32.935062,37.586178,\dots$$ is in accord with Numerical evidence suggests that all values of $t$ (the imaginary part of a ...
2
votes
1answer
289 views

Is $\frac1\pi \arctan \frac\pi{\ln x}- \frac1{\ln x}$ related to the trivial solutions $\zeta(-2n)$?

The Prime Counting Function $\pi(x)$ is given $$ \pi(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} , $$ with $ ...
23
votes
1answer
726 views

Are Primes a Self-Fulfilling Prophecy?

Assume the following process: Let's start with the set of primes $\{p_k\}$ Then we use the Euler product being equivalent to Riemann's Zeta function $$ \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = ...
1
vote
0answers
252 views

Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?

As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ...
5
votes
1answer
309 views

Square-free zeta function zeros

It is a well known fact that the geometric series $$1+x+x^2+x^3+\ldots$$ has the following form $$\frac{1}{1-x}$$ Another possible representation is $$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$ ...
8
votes
3answers
572 views

An inverse for Euler's zeta function product formula

Of course, Euler proved that the Riemann zeta function can be defined as the analytic continuation of a product over all primes. $$\zeta(s) = \prod_{p \in \mathbb{P}}\frac1{1-p^{-s}}$$ It is well ...
70
votes
5answers
5k views

What is the Riemann-Zeta function?

In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?