2
votes
0answers
44 views

The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?

Take the well known integral: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + {x}^{\frac{-s}{2}-\frac12}\right)\,\psi(x)\, ...
16
votes
2answers
415 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
0
votes
0answers
25 views

Logarithmic derivative of Riemann zeta, is this derivation correct?

Let matrix $T_2$ be defined below as the Dirichlet inverse of the Euler totient function as a function of the Greatest Common Divisor (GCD) of row index $n$ and column index $k$; $$T_2(n,k) = ...
7
votes
1answer
67 views

Riemann zeta function - Euler product formula

I want to prove that $$ \frac{1}{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}.$$ I know that the standard proof works with the Euler product formula $$\zeta(s)=\prod_{p \ \text{prime}} ...
5
votes
1answer
101 views

Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
2
votes
1answer
43 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
1
vote
1answer
68 views

Non-trivial zeros off critical line

If non-trivial zeros lay off the critical line (as shown in the picture below), would they have to come in fours rather than conjugate pairs (as the diagram shows)? I am presuming they would, ...
0
votes
0answers
25 views

Question about the zeros and poles of the PrimeZeta function.

The Euler product over all primes, $$\displaystyle \zeta(s) := \prod_{p\in\mathbb{P}} \dfrac{1}{1-\dfrac{1}{(p)^s}}$$ is only valid for $\Re(s) >1$. However, when taking the log on both sides ...
0
votes
1answer
36 views

Sum of zeta(2s) fractions without pi^(2s) in the numerators

$$ \sum _{n=1}^{\infty } \sum _{r=1}^{\infty } (\pi r)^{-2 n}=\frac{1}{2} (1-1 \cot(1)) $$ $\frac{1}{2} (1-1 \cot(1))$ is not in OEIS, so it doesn't seem to be well known. Q1: Would this info be of ...
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 ...
1
vote
0answers
49 views

Are the zeros of the sum/difference of two reflexive, entire functions all on the line $\Re(s)=\frac12$?

Remove the order $1$ pole of $\zeta(s)$ at $s=1$, to create the following entire function: $$z(s):=\zeta(s)-\dfrac{1}{s-1}$$ I like to conjecture that all complex zeros of $z(s) \pm z(1-s)$ in the ...
2
votes
0answers
95 views

A connection between a sequence from the Collatz conjecture and a sequence of densities from $\zeta(k)-1$?

Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, ...
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
56 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 \\ ...
1
vote
1answer
91 views

Relation between the prime density and Riemann's zeros?

Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers? Is there a short answer to this question, to get the overview? I once had a ...
3
votes
1answer
161 views

Elementary bound on the Riemann zeta function

I am currently preparing for a course in analytic number theory and I wanted to get a heads start. In my preparation, I came across the following problem: Show that for $|y|\geq 2$, $|\zeta(1+iy)| ...
6
votes
0answers
165 views

expressing $\log(\left \lfloor x \right \rfloor!)$ in terms of zeta-zeros

let $\psi(x)$ be the second Chebyshev Function. By the definition of this summatory function, and the fundamental theorem of arithmetic, we have the identity: $$\log(\left \lfloor x \right ...
1
vote
0answers
56 views

Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
1
vote
1answer
70 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ...
6
votes
2answers
202 views

A series related to $\zeta (3)$.

I'm not really up to date on the current status of $\zeta (3)$ but I was messing around the other day with Fourier series and found that $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3} = ...
4
votes
1answer
121 views

Why is the difference between these two infinite series equal to $\frac12$?

A follow up on this MO question: Take $s \in \mathbb{C}$, $\Re s \gt 1$, and the two infinite series: $$Z_1(s) = \sum _{n=1}^{\infty } (-1)^n \left( \frac12 + n \right) \left( {\frac ...
2
votes
0answers
103 views

Integer values of the Riemann function - II

For what value of $n \ge 2$ can we have an real $x > 0$ such that both the numbers $$ \zeta\Big(1+\frac{1}{x}\Big) \text{ and } \zeta\Big(1+\frac{1}{nx}\Big) $$ are positive integers.
3
votes
2answers
196 views

Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
9
votes
2answers
185 views

Proving a formula related with zeta function

Could you show me how to prove the following formula?$$\sum_{n=1}^\infty\frac{\zeta (2n)}{2n(2n+1)2^{2n}}=\frac12\left(\log \pi-1\right).$$ In the 18th century, Leonhard Euler proved the following ...
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: ...
0
votes
1answer
102 views

Proof of a Dirichlet's theorem using the Riemann zeta function?

Someone could tell me if there is a proof of the Dirichlet's theorem on arithmetic progressions stated below using only the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty ...
3
votes
0answers
79 views

Integer values of the Riemann Zeta function

The when $s$ is real and greater than 1, the Riemann zeta function $\zeta(s)$ takes all finite positive value $> 1$. I am studying the values of $s$ for which $\zeta(s)$ is a positive integer. I ...
3
votes
0answers
114 views

question about riemann zeta function

How can one prove that $$\zeta (2n)=\frac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}$$ where $n\in N$ and how can one prove that $$\zeta (2n)=\frac{(-1)^{n}2^{2n-2}\pi ...
4
votes
0answers
90 views

Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ I am looking for an inversion formula for the zeta ...
0
votes
0answers
44 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
2
votes
1answer
99 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 ...
4
votes
4answers
189 views

Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$

I need prove the following: ($\zeta(s)$ is the Riemann zeta function) $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I really don't know, i have tried, but nothing for now.
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 $$\log \zeta(s)=\log p\left(\sum_{n=2}^\infty \frac{1}{\log n}\frac{1}{n^s}\right)$$
3
votes
2answers
152 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
8
votes
1answer
374 views

Apéry's constant ($\zeta(3)$) value

I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
5
votes
1answer
120 views

Interesting phenomenon with the $\zeta(3)$ series

I noticed that if one takes certain partial sums of the series for $\zeta(3)$: $$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$ an interesting phenomenon occurs ...
2
votes
0answers
71 views

A function that generates 'alternating' non-trivial zeros of $\zeta(s)$

I am trying to find a function, that assuming RH, generates subsequent non-trivial zeros $\rho_n$ in an alternating way i.e.: $$\frac12+14.134...i,\frac12-21.022...i,\frac12+25.010...i, \dots$$ or ...
0
votes
1answer
83 views

Reformulation of riemann zeta

Does this extend to $\mathbb{C}$? $\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.
1
vote
1answer
66 views

What does this limit indicate?

$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$ What does this limit indicate?
12
votes
4answers
303 views

Alternating sum of multiple zetas equals always 1?

This is more in the category of "recreational math"... I was playing with multiple zetas, in the notation of $\zeta(k),\zeta(k,k),\zeta(k,k,k),\ldots$ as given in wikipedia. Looking at the alternating ...
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 ...
1
vote
2answers
159 views

Series expansion of $\zeta(s)$ using the derivatives of $\zeta(0)$

When playing with series expansions, I stumbled upon the following relation: $$\zeta(s) := \sum \limits_{n=0}^{\infty} \left( \zeta^{(n)}(0) \frac{s^n}{\Gamma(n+1)} \right)$$ that seems to hold for ...
11
votes
2answers
339 views

Can the Basel problem be solved by Leibniz today?

It is well known that Leibniz derived the series $$\begin{align} \frac{\pi}{4}&=\sum_{i=0}^\infty \frac{(-1)^i}{2i+1},\tag{1} \end{align}$$ but apparently he did not prove that $$\begin{align} ...
1
vote
0answers
93 views

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

For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
20
votes
1answer
1k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list: The Riemann Hypothesis: A Resource ...
11
votes
2answers
558 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?
4
votes
0answers
212 views

Estimation for logarithm of Riemann zeta function

Let $\sigma >1-\dfrac{c}{2\log(|t|+3)},|t|>7/8,$ where $c$ is constant from Theorem about region without zeros of Riemann zeta function. Using the fact that $$\log \zeta(s) - \log \zeta(s_1) = ...