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

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63 views

Show that $\zeta'(0)=-\frac{1}{2}\ln(2\pi)$

I started with the functional equation which was derived in class, $$ \zeta(s)=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s) $$ and took the logarithmic derivative of both sides to get $$ \...
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0answers
18 views

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=$ for Re$(z)>0$. [duplicate]

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=\frac{1}{\Gamma(z)}\int\limits_{0}^\infty \frac{t^{z-1}}{e^t+1}dt$ for Re$(z)>0$. Not sure how to get started on this, we ...
4
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1answer
92 views

Euler Product formula for Riemann zeta function proof

In class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ prime}...
3
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2answers
89 views

Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$ \frac{x}{\log x - B}. $$ (This is ...
5
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2answers
89 views

Proving that $\pi(2x) < 2 \pi(x) $

In our analytic number theory class we were given the following problem as homework: prove rigorously that for large $x$ the number of primes in $(1,x]$ exceeds that in $(x,2x]$. In class we proved ...
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0answers
45 views

How to prove that arc segment vanishes

I have this integral: $$iNx^a\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{x^{Ne^{i\theta}}e^{i\theta}}{\Gamma(Ne^{i\theta}+a-m)\zeta(2Ne^{i\theta}+2a+n)\sin (\pi \left(a+Ne^{i\theta}\right))} d\theta$...
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23 views

Generalisation of the sum operator for the divergent geometric series [duplicate]

I define a generalization of the sum operator as a linear operator from $C^N$ to C that matches with the already known operators (like the zeta regularization). With this technique, one can calculate ...
1
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1answer
36 views

Boundedness of Riemann zeta function

I've got a question. How to prove that for every $a>1$ there is $\frac{1}{a-1} \le \zeta(a) \le \frac{a}{a-1}$? Thanks!
2
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1answer
36 views

What is the relationship between all the dynamical zeta functions and the number theoretical zeta functions?

One can associate to any dynamical system a zeta function based on counting the number of fixed points of the iterates of the transformation. Explicitly we have: $$\zeta_{A} = exp \left( \sum_{n=1} \...
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2answers
113 views

How do you use the Riemann Zeta Function?

I know that the Riemann Zeta Function is defined as: $$\zeta (s)=\sum_{n=1}^\infty \frac {1}{n^s}=\frac {1}{\Gamma (s)} \int _0^{\infty}\frac { x^{s-1}}{e^x-1} dx$$ Which I think would prove useful ...
10
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1answer
114 views

Equality involving Hasse zeta function of commutative ring finitely generated over $\mathbb{Z}$

Let $\mathbb{F}_q$ be a finite field consisting of $q$ elements. Imitating Riemann's zeta function$$\zeta(s) = \sum_{n = 1}^\infty {1\over{n^s}},$$define$$\zeta_{\mathbb{F}_q[t]}(s) = \sum_f {1\over{\...
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0answers
41 views

Convergence of this integral related to Riemann-Zeta Function

Is it possible to show that for $Re(s)>0$ the following integral converges. $$\displaystyle\int_1^\infty \left[{x^{\frac{s}{2}-1} + x^{-\frac{s+1}{2}} }\right] \omega \left({x}\right)dx$$ Where $\...
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50 views

Recurrence for the Mertens function generalized to complex numbers.

It is well known that the sum of the Möbius function $\mu(n)$ over divisors is zero unless $n=1$. $$\sum\limits_{d|n} \mu(d) = \delta_{n \, 1}$$ where $$\delta_{n \, 1}$$ is Kronecker delta. Or put ...
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44 views

Fundamental theorem of arithmetic, prove that these matrices are the same apart from the main diagonal.

I know and I can prove that the fundamental theorem of arithmetic is encoded uniquely by this recurrence written in Mathematica 8: ...
4
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2answers
136 views

Analytic continuation for $\zeta(s)$ using finite sums?

$\zeta(s)$ converges for $\sigma >1$ but not for $\sigma =1/2.$ But for some reason for $s = 1/2 + i t $ and fixed finite $N,~$ $\zeta_N(s) =\sum_{n=1}^N\frac{1}{n^s}$ is very close to $\zeta(s)$ ...
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31 views

$\eta(s)+\eta(1-s)=F(s)-G(s)$ and roots of $F(s),G(s)$ are on the critical line

Wusheng Zhu in 2012 uploaded to arxiv.org an interesting preprint titled "Riemann Zeta Function Expressed as the Di fference of Two Symmetrized Factorials Whose Zeros All Have Real Part of 1/2" (arxiv:...
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1answer
46 views

Series involving Riemann Zeta Functions

I want to know if there exists any result about the exact or an approximate value of the following sum of the infinite series involving Riemann Zeta functions. Any pointer towards related results will ...
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0answers
37 views

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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1answer
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 ...
2
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1answer
72 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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1answer
40 views

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, $$\max_{z\...
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23 views

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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30 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 $$\int_{x^{1/n}}^\...
6
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2answers
150 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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0answers
123 views

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 - d+it\right)\...
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3answers
89 views

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 (2n)...
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37 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
97 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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4answers
167 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 $n\geq 1$...
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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
61 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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46 views

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 $$ b_{j}=(j+1/4)\ln n\tag{1}$$ $$c_{j}=\frac{(2j+1)\pi^j}{2\Gamma(j)} \sum_{k=1}^{n}k^{2j}\tag{2}$$ $$w_{a}(x)=u_{a}...
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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 $$\psi(x)=\...
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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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1answer
48 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 ...
5
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1answer
132 views

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 $\xi$-...
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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 half-...
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2answers
56 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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4answers
77 views

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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2answers
65 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 $\gcd(...
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1answer
43 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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0answers
46 views

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 ...
1
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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 1}\left(\zeta(x)-\frac{1}{x^x-1}\...
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1answer
36 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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0answers
27 views

On a simplification of zeta functions.

Can you simplify the following product? With the known properties of $\zeta(x)$ $$\color\green {\frac{\zeta\left(\frac{1}{2}\right)\zeta\left(\frac{1}{3}\right)\zeta\left(\frac{1}{4}\right)}{\zeta\...
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1answer
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$: ...
2
votes
3answers
207 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 $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+..>\frac{1}{1}+\frac{1}{2}+\...