Questions on the famed $\zeta(s)$ function of Riemann, and its properties.
1
vote
1answer
225 views
about the riemann zeta function and the prime counting function
i have posted this question on MO, and they referred me to post here .
one starts with the formal definition of zeta :
$$\displaystyle \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} $$
then :
$ \ln(\zeta ...
24
votes
1answer
844 views
Are these zeros equal to the imaginary parts of the Riemann zeta zeros?
The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$:
$$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot e^{-\frac{x}{2}})$$
can be plotted with ...
4
votes
1answer
150 views
Zeta function identity
How does one prove the zeta function identity
$$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
7
votes
2answers
599 views
Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$? [closed]
Why is $\zeta(2) = \frac{\pi^2}{6}$ almost equal to $\sqrt{e}$?
Experimenting a bit I also found $\zeta(\frac{8}{3}) \approx e^\frac{1}{4}$, $\zeta(\frac{31}{9}) \approx e^\frac{1}{8}$ and ...
2
votes
2answers
228 views
zeta of three, question about closed form
If $\sum\limits_{n=2}^\infty \frac1{(n^2-n)^3}=10-\pi^2$, then what is the limit in closed form of $\sum\limits_{n=1}^\infty \frac1{n^3}$?
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0answers
171 views
What are the local minima in this spectrum?
Edit 6.2.2012: The sequence to be transformed should be f = 0,1,2,3,4,5... which makes the mentioning of the von Mangoldt function less necessary.
Edit 5.2.2012: I had the wrong plot of the insignal. ...
3
votes
3answers
266 views
Simplest proof that $\zeta(s) \to \infty$ as $s \to 1$?
For homework I had to prove the divergence of the series $1/(k\log^p k)$ for all real $p$ (it is simple to do so via integration.) However a more elegant means would be to appeal to the behavior of ...
1
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1answer
197 views
Verifying identities for Riemann zeta function
I ran across these two problems, while reading a text on number theory. The problem states: "Verify the following identities". What does "verify" mean in this context, and what strategies can I employ ...
5
votes
3answers
279 views
Limit of Zeta function
I'm looking for a reference for (or an elementary proof of)
$$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$
Thanks for your help.
2
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1answer
156 views
A question from Titchmarsh's Riemann Zeta Function textbook.
I have one query, concerning the newest edition of this monograph.
At page 7, section 1.2, at the bottom of the page, it's written that:
" It is easily seen that $\zeta(s)=2$ for $s=\alpha$, where ...
19
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1answer
649 views
Are Primes a Self-Fulfilling Prophecy?
Assume the following process:
Let's start with the set of primes $\{p_k\}$
Then we use the Euler product being equivalent to Riemann's Zeta function
$$
\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = ...
19
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1answer
452 views
Upper bound on differences of consecutive zeta zeros
The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
2
votes
1answer
276 views
Primes and Riemann zeta function.
Primes numbers and Riemann zeta function.
Question 1: Is there a proof of the infinitude of prime numbers using the Riemann Zeta function. Exboço could show me a proof of this where I could find it?
...
2
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1answer
149 views
Dirichlet character over Riemann zeta function
Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
8
votes
1answer
138 views
Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.
Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
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0answers
143 views
Has it been ruled out that the Riemann hypothesis fails for only finite number of zeros?
Has it been ruled out that the Riemann hypothesis fails, but fails only for finite number of zeros?
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1answer
64 views
Another question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$
This is a question distinct from but related to the question I wrote here: Question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$, so assume the same things that I wrote there.
The paper then ...
4
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1answer
136 views
Question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$
I'm following this paper: http://mathdl.maa.org/images/upload_library/22/Chauvenet/Zagier.pdf
Define $\Phi(s) = \displaystyle\sum_p \frac{\log p}{p^s}$.
By taking a logarithm and differentiating ...
3
votes
3answers
486 views
Order of growth of $(s-1)\zeta(s)$
Again, order of growth problems.
Show that the function $(s-1)\zeta(s)$ is an entire function of growth order $1$; or equivalently,
$$|(s-1)\zeta(s)| \leq A_{\epsilon} \; \exp ...
3
votes
2answers
162 views
Approximate Riemann zeta function
Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.
In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.
My question is: Is there a Functional equation for ...
5
votes
0answers
169 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 ...
12
votes
1answer
287 views
challenging integral involving $\zeta(5)$
I ran across a curious integral that seems to be rather tough that some on the site may enjoy.
Show that $$\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^{2}}}{1-x^{2}\sin^{2}(x)}dx = ...
5
votes
2answers
377 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
2
votes
1answer
202 views
Why are Gram points for the Riemann zeta important?
Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a Gram point and the imaginary part of the zeros (under the ...
7
votes
2answers
330 views
How to find $\zeta(0)=\frac{-1}{2}$ by definition?
I would like to know how we can find the following result:
$\zeta(0)=\frac{-1}{2}$
Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$
to find this?
1
vote
0answers
165 views
Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?
As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ...
13
votes
3answers
1k views
The main attacks on the Riemann Hypothesis?
Attempts to prove the Riemann Hypothesis
So I'm compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible ...
4
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0answers
175 views
New generalization of Riemann Zeta?
I am interested in the following generalization of the Riemann Zeta function:
$$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$
This is most closely related ...
2
votes
0answers
195 views
Why Pólya's method is wrong?
By truncating the Fourier transform, Pólya managed to prove that the Xi function on the critical line was approximately
$$\xi(1/2+is) = (2\pi)^2 ( K_{9/4+is/2}( 2\pi) +K_{9/4-is/2}( 2\pi))$$
If this ...
1
vote
1answer
116 views
Can you provide a lower bound on $|\zeta (s) |$ for fixed $\mathrm{Re}(s) > 1$?
It's easy to prove, for example, that $|\zeta(2 + it)| > 2 - \frac{\pi^2}{6}$. However, there is some $\sigma > 1$ for which $\zeta ( \sigma ) = 2$, and it is more difficult to obtain a lower ...
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votes
2answers
309 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
11
votes
2answers
601 views
Logarithmic derivative of Riemann Zeta function
Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$?
I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes
...
5
votes
2answers
515 views
Why does the Riemann zeta function have zeros in the complex plane? How is it possible to find them?
I ask this because, according to Euler's product formula, Riemann's zeta function =(1/something), so how could that be zero?
Also, how could one find zeros that are on the negative side and find a ...
1
vote
1answer
410 views
How is $\zeta(0)=-1/2$? [duplicate]
Possible Duplicate:
Why does $1+2+3+\dots = {-1\over 12}$?
Fermat's Dream by Kato et al. gives the following:
$\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}$ (the standard Zeta ...
7
votes
1answer
221 views
Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$
It seems that the $n$th cumulant of the uniform distribution on the interval $[-1,0]$ is $B_n/n$, where $B_n$ is the $n$th Bernoulli number.
And also $-\zeta(1-n) = B_n/n$, where $\zeta$ is Riemann's ...
8
votes
1answer
212 views
An infinite series involving the Zeta Function
I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants:
$$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$
and
...
5
votes
2answers
316 views
Order of Zeros of the Riemann Zeta Function
Is it true that all zeros of the Riemann Zeta Function are of order 1?
Let $h(z) = \frac{\zeta'(z)}{\zeta(z)}\frac{x^z}{z}$, where $x$ is a positive real number ($x > 1$, probably?) , and $\zeta$ ...
10
votes
2answers
344 views
Do these series converge to logarithms?
It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$
If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$
Starting:
...
3
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0answers
94 views
Order of growth of real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$
On a lark, I decided to calculate (via Newton's method and using mpmath) the real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$ for as many $n\in\mathbb{N}$ as I could. What sort of surprised me is ...
10
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1answer
316 views
elliptic generalizations of Euler's trick
So Euler employed the following identity
$$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$
to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$
I'm curious if there's been ...
2
votes
1answer
96 views
Multiplication of coefficients in Dirichlet series
This appears to be a relationship:
$\sum\limits_{p\;\text{prime}} \frac{1}{p^s} = \log\zeta (s) - \sum\limits_{n=1}^{\infty}\frac{\sqrt{a_{n}b_{n}}}{n^{s}}$
where $a_{n}$ is a sequence of fractions ...
12
votes
3answers
1k views
Analytic continuation- Easy explanation?
Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said,
The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
4
votes
0answers
283 views
Is this a relation between the Riemann zeta function and the Prime zeta function?
I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function?
$$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$
...
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0answers
129 views
Determining well-definedness for functions
How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function?
Firstly:
$$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$
...
34
votes
5answers
3k views
Why does $1+2+3+\dots = {-1\over 12}$?
$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$
$\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ ...
8
votes
2answers
387 views
Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?
The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 ...
2
votes
0answers
74 views
L-function of product
Given two varieties of finite type over a finite field, what is the $L$-function of their product in terms of the $L$-function of the factors?
10
votes
2answers
361 views
Integrating $\frac{x^k }{1+\cosh(x)}$
In the course of solving a certain problem, I've had to evaluate integrals of the form:
$$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$
for several values of k. I've noticed that that, for k a ...
6
votes
1answer
185 views
What is known about the pattern for $\zeta(2n+1)$?
Related to the question Does $\zeta(3)$ have a connection with $\pi$?:
It is well known that
$$\zeta(2n) = f(2n) \pi^{2n}$$
where $f(n)$ is an function in rationals: (the denominator = OEIS ...
28
votes
5answers
2k views
Does $\zeta(3)$ have a connection with $\pi$?
The problem
Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)?
Details
Several ...
