Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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16
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88 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
12
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472 views

I can Euler-sum $\sqrt{-\ln(1)}-\sqrt{-\ln(2)}+\sqrt{-\ln(3)}-\cdots$. But how can I do $\sqrt{-\ln(1)}+\sqrt{-\ln(2)}+\sqrt{-\ln(3))}+\cdots$?

This is also related to an older thread in MSE ("what is the half derivative of zeta at zero?") . One of the possible steps in the problem of that thread was to evaluate the series ...
10
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107 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
9
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253 views

Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get ...
9
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178 views

A cotangent series related to the zeta function

$$\sin x = x\prod_{n=1}^\infty \left[1-\frac{x^2}{n^2\pi^2}\right]$$ If you apply $\log$ to both sides and derivate: $$\cot x = \frac{1}{x} - \sum_{n=1}^\infty \left[\frac{2x}{n^2\pi^2} ...
9
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230 views

Integer values of the Riemann Zeta function

When $s>1$ is real, 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 have the following ...
9
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468 views

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

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
9
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1k views

Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
8
votes
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352 views

New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$ This is most closely related ...
7
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77 views

Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as ...
7
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136 views

An inequality for $|\zeta (s,a)|$, a detailed proof

In page 272 of [1], Apostol leaves as a reader's assigment to complete a proof of a related statement with Hurwitz zeta function, defined initially for $\sigma >1$ by the series ...
7
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213 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 ...
6
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211 views

One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the ...
6
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213 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
6
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92 views

Why is $\zeta(s)\neq0$ for $\operatorname{Re}(s)=0$?

I have a question concerning the Riemann zeta function for a project I've been working on. Why is it that $\zeta(s)\neq0$ for $\operatorname{Re}(s)=0$ (that is, there are no non-trivial zeroes of the ...
6
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200 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 ...
6
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234 views

On the series $ \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) $.

Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it? $$ \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
6
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135 views

Why should the eigenvalues of random matrices reflect zeta function zeroes?

As per this article. And also : Is this a particular property of 2 dimensional objects? Could random vectors also model the "universality phenomena" -- globally random distribution of zeroes combined ...
6
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332 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
6
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405 views

Is this a relation between the Riemann zeta function and the Prime zeta function?

I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function? $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ ...
5
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55 views

zeros/poles of Laplace transforms of Dirac combs (Riemann zeta function)

let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle ...
5
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88 views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
5
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65 views

Can this approximate closed form of Apery's constant $\zeta(3)$ be improved?

I know that an approximate closed form is not really a solution. However, I would like to present a method that gives a closed form of $\zeta(3)$ that is accurate to the 5th decimal, hoping that it ...
5
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118 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
5
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114 views

Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
5
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80 views

English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann $\xi$ function

I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung über die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48(1926), 305-317; ...
5
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135 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
5
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96 views

Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
5
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101 views

Is there a name for this constant?

$$\prod_{n=2}^\infty \zeta(n)=2.294856591673313794183$$ Is there a name for this constant and what are some if its properties?
5
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130 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 ...
5
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275 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) = ...
5
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1k views

Zeta function values in terms of Bernoulli numbers.

The material presented at this link on Zeta function values at even integers proposes a method to compute these that is based on Euler's work. I would like to present a short proof for your ...
4
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79 views

how to show this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = H \left( - \frac{i}{2} (2 t - 1) \right) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left(\frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right)\right) ...
4
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206 views

Approximate zeros of a (hypothetical) analog of $\zeta(s)$

[Added numbers 11/13.] Motivation (can skip). When prime powers $p_n$ are used to calculate $$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$ for (say) $N= 30,$ $x>5$, at ...
4
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117 views

riemann zeta function : entire and even Laplace transforms

$$\xi(s) = s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s)$$ $$\xi(s) = \xi(1-s)$$ thus $\Xi(s) = \xi(1/2+s) = \Xi(-s)$ is even, and furthermore it is an "entire and even Laplace transform" : $$\Xi(s) = ...
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75 views

Zeta zeros standard normal distribution

Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for ...
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71 views

Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
4
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174 views

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: ...
4
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103 views

An infinite series that gives $f(s)=s$. How could it be explained more easily?

This question loosely builds this one. Equate the following two infinite series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{4\,(s-1)} \left(1+s+\sum _{n=1}^{\infty } \left( {\frac ...
4
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120 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
4
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105 views

Riemann Zeta Function Analytic Continuation

I am struggling to understand how the analytic continuation of the Riemann Zeta function is derived to extend it to all complex values $z$ not equal to $1$, starting with the series which converges ...
4
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180 views

An interesting identity involving powers of $\pi$ and alternating zeta series

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
4
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121 views

Is this the chord G Major I am hearing as base tones from interference of zeta zeros times eigenvalues of the von Mangoldt function matrix?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
4
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211 views

Is this formula for $\zeta(15)$ true?

Apery gave, $\begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned}$ J. Borwein and D. Bradley found this can be generalized to ...
4
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111 views

Order of growth of real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$

On a lark, I decided to calculate (via Newton's method and using mpmath) the real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$ for as many $n\in\mathbb{N}$ as I could. What sort of surprised me is ...
3
votes
0answers
49 views

If these two expressions for calculating the prime counting function are equal, why doesn't this work?

So I've seen some different explanations of how the zeros of the zeta function can predict the prime counting function. The common example is that $$\pi(x)=\sum_{n=1}^\infty ...
3
votes
0answers
61 views

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line?

What are the (possible) fundamental reasons for all zeros of zeta function to be on the critical line ? Are there particular mechanism to make this happen ? Because of Levinson and Corney's work, we ...
3
votes
0answers
48 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
3
votes
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140 views

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
3
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0answers
139 views

Riemann vs. Ihara's $\zeta$ Function Variable Question

The Euler product for the Riemann zeta function $\zeta(z)$ implies that $$ \log\zeta_R(z)=\sum_{m>0}\frac{P(mz)}{m} \tag{R}, $$ whereas the Ihara zeta function for a graph $G$--all can be ...