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

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2
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23 views

Significance of the Riemann hypothesis to algebraic number theory?

Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? ...
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31 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
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2answers
69 views

An integral related with the Riemann $\zeta$ function

I have to prove that: $$ \forall s>1,\qquad\int_0^\infty \sum_{k=1}^{\infty}\frac{1}{(k^s+1)^x+k^s}dx=\zeta(s). $$ I how do I find the closed form for this sum? $$ ...
0
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1answer
41 views

Riemann Zeta Function for $\Re(s)=0$

All the sources I have read talk about continuation from $Re(s)>1$ to $Re(s)>0$ then $Re(s)<0$ $(s\neq 1)$. What about $Re(s)=0$? Where does that go?
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1answer
57 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
16 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
75 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{ ...
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2answers
85 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
84 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
40 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))} ...
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0answers
17 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
vote
1answer
35 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!
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1answer
35 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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0answers
42 views

Give an expression for the probability that k randomly chosen natural number share a common factor

I know the probability is $\prod (1-p^{-k}) = \frac{1}{\zeta(k)}$, and $\zeta(k) = \sum_{n=1}^{\infty}\frac{1}{n^{k}} = \frac{1}{1^{k}} + \frac{1}{2^{k}} + \frac{1}{3^{k}} + \frac{1}{4^{k}} + \cdots$. ...
2
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2answers
101 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 ...
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1answer
108 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 ...
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0answers
37 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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0answers
47 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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0answers
41 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
votes
2answers
133 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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27 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" ...
2
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1answer
45 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
votes
1answer
69 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 $\text{Re}(s) \leq 1$. ...
0
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1answer
38 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, ...
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22 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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0answers
35 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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0answers
26 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 ...
4
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2answers
132 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
118 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 - ...
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3answers
88 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 ...
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0answers
35 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
82 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
160 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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0answers
42 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
55 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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43 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 $$ b_{j}=(j+1/4)\ln n\tag{1}$$ $$c_{j}=\frac{(2j+1)\pi^j}{2\Gamma(j)} \sum_{k=1}^{n}k^{2j}\tag{2}$$ ...
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1answer
50 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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0answers
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
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 ...
5
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1answer
121 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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0answers
26 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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40 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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vote
1answer
50 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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vote
2answers
49 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
74 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 ...
1
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2answers
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 ...
0
votes
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 ...
1
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0answers
43 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 ...