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

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Are there iterative formulas to find zeta zeros?

I am wondering whether one could find Riemann zeta zeros iteratively by using relationships such as this one: $$\rho _1=\lim_{s\to 1} \, \frac{\zeta (s) \zeta \left(s \cdot \rho _1\right)}{\zeta ...
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38 views

Does every divergent series $\sum_{n=1}^{\infty} a_n$ (with $a_n>0$) grow like $\int a_n dn$ in the limit $n \to \infty$?

From the definition of Euler's constant I knew that the following is true $$\lim_{N \to \infty} \left( \sum_{n=1}^{N} \frac{1}{n}-\log N \right)=\gamma$$ I decided to check for other similar ...
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71 views

Mobius function and exp(2 pi i n x)

We know that \begin{equation} \sum_{n \geq 1} \frac{\mu(n)}{n^{s}} = \frac{1}{\zeta(s)}, \end{equation} and so, the left series can be plainly analytically continued to $\text{Re}(s) \leq 1$. ...
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How can we compute the maximum of $|\zeta(z)|$ on the square with vertices $2,3,3+i$ and $2+i$?

By the maximum modulus principle, since $\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^z}$, with $\Re z>1$, is analytic inside the square Q of vertices $2,3,3+i$ and $2+i$, and continuous on Q, ...
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Residue for $\frac{\zeta(s)}{\zeta(2s)}$ at zeros of $\zeta(2s).$

I want to calculate residue at the poles for $\frac{\zeta(s)}{\zeta(2s)}.$ For pole of numerator $s=1$ I have calculated the residue. I am having trouble at the zeros of denominator. Basically I am ...
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40 views

Is the Riemann Zeta function negative in the critical strip

Just saw a question on "how to prove that the Riemann Zeta function is negative in the critical strip". What is meant by Zeta(s) < 0? Does it mean that it's real part is negative, or both real and ...
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27 views

Integral in Prime counting function

I've come across on another stack exchange question somewhere that gave a definition for the integral that appears in Riemann's prime counting formula. The integral in question is ...
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140 views

Why isn't there a formula for $\zeta(k)=\sum_{n=1}^\infty\frac{1}{n^k}$ involving $\pi$ when $k$ is odd?

Now we know that $\sum \frac{1}{n}=\text{divergent}, \sum \frac{1}{n^2}=\frac{\pi^2}{6}$ but now this for $\sum \frac{1}{n^3}=1.20....$ and again $\sum \frac{1}{n^4}=\frac{\pi^2}{90}$ .Now somewhere ...
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In Pursuit of the Modulus of the Riemann Zeta Function in the Critical Strip

Notation: $\zeta(s) = \zeta(x+it)$ In http://dml.cz/bitstream/handle/10338.dmlcz/136881/MathSlov_53-2003-2_3.pdf the following inequality was proven: $$ \left|\zeta\left(\frac 1 2 - ...
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Asymptotic formula for $\prod_{k=1}^{\infty}\zeta (2kn)$

Suppose $n\geq 1$ is a positive integer. Can we find an asymptotic formula for this product below. $$\prod_{k=1}^{\infty}\zeta (2kn)=\zeta (2n)\zeta (4n)\zeta (6n) \cdots$$ I tried to use $\zeta ...
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36 views

Who extended the Euler Product Formula to all real $s>1$?

I believe Euler discovered this identity but only wrote them for particular values of $s$, then Chebychev extended to real $s>1$. However, I read in the book Riemann's Zeta Function, H.M. Edwards ...
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1answer
96 views

How many elementary ways are there to prove that $\displaystyle\left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $?

In a comment under this answer, a user boldly asserts that there is ONLY ONE way to prove that $$ \left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $$ where $\zeta$ is Riemann's zeta function. ...
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163 views

How to show $\zeta (1+\frac{1}{n})\sim n$

How to show $\zeta (1+\frac{1}{n})\sim n$ as $n\rightarrow \infty$ where $\zeta$ is the Riemann zeta function. And can we say $\lceil \zeta (1+\frac{1}{n}) \rceil=n$ for any positive integer $n\geq ...
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43 views

For the Riemann Zeta Function, is there a relation between modulus[Zeta(x+it)] and modulus[Zeta(it)]?

Graphs of numerical examples show that in the critical strip: modulus(Zeta(x+it) < modulus[Zeta(it)]. The question concerns the possibility of generalizing this inequality. In ...
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1answer
60 views

Riemann zeta-function functional equation form

In Titchmarch's book the functional equation is given as $$\zeta(s)=2^s\pi^{s-1}\sin \left( \frac{s\pi}2\right) \Gamma(1-s)\zeta(1-s).$$ However, in the third proof, he derives a following equation ...
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How to prove $w_2(x)>w_1(x)$?

Let $x\in\mathbb{R},x\geq 0$;$m,n\in\mathbb{N};n>2,m>>n$;$a=1,2$ and let $$ b_{j}=(j+1/4)\ln n\tag{1}$$ $$c_{j}=\frac{(2j+1)\pi^j}{2\Gamma(j)} \sum_{k=1}^{n}k^{2j}\tag{2}$$ ...
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1answer
58 views

Riemann zeta-function functional equation proof

I'm reading through Titchmarch's "The Theory of the Riemann Zeta-Function" and there's a part in the functional equation proof number 3 that I haven't figured out. He defines a function ...
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34 views

speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$

I would like to find the speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$ for any $z$. For example, if $|z| < 1$, doing taylor expansion we know that it is roughly $e^{zn}.$ But I need it ...
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47 views

When $\frac{\pi ^{x}}{\zeta (x)}$ is rational?

When $n$ is a positive integer, we know $$\zeta (2n)=\frac{(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}$$ Now let's say $x>1$ is a real number. Can we say if $\frac{\pi ^{x}}{\zeta (x)}$ is a rational ...
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Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz $

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds $$ It looks like a complicated ...
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28 views

Growth rate of $\zeta(n)^{-1}$

What is the asymptotic growth rate of $\frac1{\zeta(n)}$? Is it polynomial in $n$?
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42 views

Is the absolute value of the intersection of two functions related to the nontrivial zeros always equal to $\sqrt{2}$?

With $\displaystyle \chi(s)=\pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)$ and $K(s)=\Psi\left(\frac{s}{2}\right)-\ln(\pi)$, with $\Psi\left(s\right)$ the digamma function, then the Riemann ...
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1answer
51 views

Root of the $\zeta(s) = s$

What is the root $s_0$ of the equation $\zeta(s) = s$, where $\zeta(s)$ is Euler zeta function? This point $s_0$ has obvious property: the segment $(1,s_0]$ to the left of it is mapping on the ...
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2answers
55 views

Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$

Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$. What happens to the zeta function at these points? For example $\sum_{n=1}^\infty \frac1{n^s}$ is defined for $\Re(s)>1$ and for $\Re(s)>0$ ...
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How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?

$\lim_{s \rightarrow \infty} \zeta(s) = 1$ I have seen a proof using the fact $1 \leq \zeta(s) \leq \frac{1}{1-2^{1-s}}$ but this relies on proving the inequality first which is quite cumbersome. I ...
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62 views

Is this true :$\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$?

I would like to give other representation for zeta function using fundemental arithmitic I have got this: $\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$ where ...
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1answer
42 views

Integrating $ x^{\frac{3}{2}} \frac{1}{1 + e^x} $

I'm wondering if this integral can be expressed in some compact form: $$ \int\limits_{0}^{\infty} x^{\frac{3}{2}}\frac{1}{1 + e^x}dx $$ And if not - why? I was thinking that it was somehow ...
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Euler Product Formula - Zeta Function

For $s \in \mathbb{C}$ and $\sigma = \Re(s)>1$, $$\zeta(s) = \prod\limits_{p \in \mathbb{P}}\left(1 - \frac{1}{p^s}\right)^{-1}$$ My question is: is the above correct? Or should the $s$ be ...
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1answer
63 views

What about $\lim_{x\to 1}\left(\zeta(x)-\frac{1}{x^x-1}\right)=1+\gamma$?

When I type 1 in the box lim x to, and zeta(x)-1/(x^x-1) in the box Function, of this online calculator (Wolfram Alpha) one has as output $$\lim_{x\to ...
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1answer
31 views

What are the equivalent statements of GRH using the Möbius or Liouville functions?

We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds. Are there similar ...
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On a simplification of zeta functions.

Can you simplify the following product? With the known properties of $\zeta(x)$ $$\color\green ...
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12 views

What is the meaning of cyclically equivalence classes of multiple indices?

Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set? $$ I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$ where $ w$: ...
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3answers
206 views

Why $\zeta (1/2)=-1.4603545088…$?

I saw $\zeta (1/2)=-1.4603545088...$ in this link. But how can that be? Isn't $\zeta (1/2)$ divergent since ...
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1answer
142 views

Finding a solution to $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$

Finding ONE solution to: $$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$ can apparently be done by iterating the following formula: $$\Large s(m+1)=\frac{\log \left(-\frac{1}{\sum _{n=1}^{k-1} ...
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We derive the nontrivial zeta zeros from the primes - can we use the same method to derive a set from the zeros, and in general for some set {S}?

The number of primes less than a given $x$ have an asymptotic formula and from that, we get a pretty good approximation. The error term between this approximation and the actual value comes from the ...
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A problem on simplification $\operatorname{Li_3}\frac{1}{3}$

Can you simplify $$\large\operatorname{Li_3} \frac{1}{3} $$ It might be impotant to note that $$\operatorname{Li_3}\frac{1}{2}=\frac{7\zeta(3)}{8}+\frac{\log^3 2}{6}-\frac{\pi^2\log 2}{12}$$ But I ...
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Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
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201 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 ...
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Did Hardy prove that there are countably, or uncountably many zeros on the line Re$(s)=1/2$ of $\zeta(s)$?

It's known that Hardy proved that there are infinitely many zeros of $\zeta(s)$ on the line Re$(s)=\frac{1}{2}$, but did he prove it's countably infinite? Or uncountable?
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can this functional of the Hardy Z function be written as an elliptic theta function?

Can H(t), Equation 33 of http://vixra.org/pdf/1510.0475v7.pdf be expressed as an elliptic theta or related function ? $H(t)= {\frac {4\,i\zeta \left( -i/2 \left( i-2\,t \right) \right) \pi \, ...
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Modular Euler product?

We know the Euler product. $$\zeta (s)=\prod_{p}\frac{p^{s}}{p^{s}-1}$$ I wonder if there is formula or any kind of work for this kind of prime product below? $$\prod_{p\equiv a \ (mod \ ...
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28 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
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48 views

An Analogous Riemann Integral

$$1=\sum_{n=2}^\infty (\zeta (n)-1)$$ is a fairly well known result W|A validates this result Is there a closed form to the analogous integral: $$\text{?}=\int_2^\infty \text{d}x \, (\zeta(x)-1)$$ ...
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75 views

Why are the trivial zeros of the Riemann zeta function only negative?

The functional equation of the Riemann zeta function is $$\zeta(s)=2^s\pi^{s-1}\sin(s\pi/2)\Gamma(1-s)\zeta(1-s)$$ clearly $2^s$ and $\pi^{1-s}$ are never equal to zero on the complex plane, and ...
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92 views

Difficult expression of sum

I wanna show that $\sum{\frac{1}{k^{4}}}=\frac{\pi^{4}}{90}$. For this, I know that $$\sin(z)=z-\dfrac{z^{3}}{3!}+\dfrac{z^{5}}{5!}-\dfrac{z^{7}}{7!}+\cdots$$ On the other hand, also know that ...
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139 views

Wouldn't the Riemann hypothesis rule out a formula to predict primes? [closed]

Prime formula: a deterministic way to predict primes. Riemann hypothesis: implies "primes are random". If RH is true will we never have a useful prime formula?
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132 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 ...
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1answer
40 views

Convexity of the Riemann-zeta function without derivative

Proving that the $\zeta$ function is convex on $(1,+\infty)$ is pretty simple if we use the derivative, but is there a proof without using derivative? I'm allowed to use just the definition of the ...
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1answer
78 views

Riemann zeta and Dirichlet eta functions, and Cauchy-Riemann equations

Taking the complex argument of the complex number $2$ as $0$, I've computed for complex numbers $s=x+iy$ $$1-2^{1-s}=1-2^{1-x}(\cos(y\log 2)-i\sin(y\log 2)),$$ in the equation ...
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173 views

Product of two series to get a series decomposition of zeta in the critical strip

$\def\sfrac#1#2{% \small#1% \kern-.05em\lower0.1ex/\kern-.025em% \lower0.4ex\small#2}$I've been working on gaining an intuitive understanding of the analytic continuation of the zeta ...