Questions on the famed $\zeta(s)$ function of Riemann, and its properties.
1
vote
0answers
21 views
The partial sum and partial product of $\zeta$function
Taking the partial sum of the $\zeta $ function:
$$\zeta^H(s,k)=\sum_{n=0}^k \frac{1}{n^s}$$
and the partial product f the $\zeta $ function:
$$\zeta^P(s,j)=\prod_{i=0}^j \frac{1}{1-p_i^{-s}}$$
I ...
13
votes
3answers
132 views
Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$
I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
4
votes
1answer
73 views
question about Riemann zeta $\zeta (0)$
i know that
$$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$
so
$$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty $$
but actually
$$\zeta (0)=-0.5$$
where is the wrong
please help
thanks ...
8
votes
1answer
59 views
product of zeta(k)
The product $\prod_{k\geq 2}\zeta(k)$ converges. Is the limit a known constant ?
(This infinite product is involved in the estimation of the covolume of $SL_n(\mathbf Z)\backslash SL_n(\mathbf ...
4
votes
3answers
168 views
Factorial of infinity
So, I've read in this article that:
$$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$
And that:
$$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$
I found this result very ...
0
votes
4answers
74 views
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]
Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
0
votes
0answers
27 views
Is there a pair correlation function for primes?
Montgomery's pair correlation function for the non-trivial zeros of the Riemann $\zeta(s)$ function is defined via the term $$1- \left( \frac {\sin(\pi u)}{\pi u} \right)^2$$
Does anybody know if ...
6
votes
1answer
140 views
Apéry's constant ($\zeta(3)$) value
I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
5
votes
1answer
105 views
Interesting phenomenon with the $\zeta(3)$ series
I noticed that if one takes certain partial sums of the series for $\zeta(3)$:
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$
an interesting phenomenon occurs ...
3
votes
1answer
93 views
Zeta function zeros and analytic continuation
I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemman Zeta ...
1
vote
2answers
46 views
Proof that the zeta function converges for Re(s)>1
It would be absolutely fantastic if anybody could give me some guidance on the question above. For me (please correct me if I'm wrong), this question boils down to proving that
...
3
votes
0answers
118 views
Identity involving $\zeta(3)$
This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that
...
1
vote
0answers
42 views
A function that generates 'alternating' non-trivial zeros of $\zeta(s)$
I am trying to find a function, that assuming RH, generates subsequent non-trivial zeros $\rho_n$ in an alternating way i.e.:
$$\frac12+14.134...i,\frac12-21.022...i,\frac12+25.010...i, \dots$$
or ...
3
votes
1answer
72 views
What is the half-derivative of zeta at $s=0$ (and how to compute it)?
I'm trying to understand the concept of fractional derivatives and am fiddling with the examples at wikipedia. The a'th derivative of a monomial in x, where a can be fractional is accordingly $$ {d^a ...
0
votes
1answer
56 views
Reformulation of riemann zeta
Does this extend to $\mathbb{C}$?
$\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.
9
votes
2answers
248 views
Clausen and Riemann zeta function
This is an exercise from the American Monthly Problems from last year.
I would like prove two formulas:
(1) $\int_0^{2\pi}\int_0^{2\pi}\log(3+2\cos(x)+2\cos(y)+2\cos(x-y)) dxdy=8\pi ...
1
vote
1answer
63 views
What does this limit indicate?
$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$
What does this limit indicate?
2
votes
1answer
52 views
What's the lowest real $x$ such that $\zeta(x)$ converges?
It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$
diverges, and
$$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$
converges to $\frac{\pi^2}{6}$.
Intuiting the ...
11
votes
4answers
211 views
Alternating sum of multiple zetas equals always 1?
This is more in the category of "recreational math"...
I was playing with multiple zetas, in the notation of $\zeta(k),\zeta(k,k),\zeta(k,k,k),\ldots$ as given in wikipedia.
Looking at the alternating ...
0
votes
2answers
101 views
Quantum uncertainty can explain the Riemann Hypothesis?
In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
0
votes
1answer
108 views
Can anyone provide a proof for this conjecture?
Theorem?:
Let $n$ be a positive interger, $n>1$, then Riemann zeta function can be expressed in terms of a multiple integral which exhibits the following form:
$$ \displaystyle
...
9
votes
1answer
87 views
A Tough Series $\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$
I have done series with $\zeta(2k)$ and $\zeta(k)$, but I have no idea with this one:
$$\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$$
This value was given by Mathematica. Any hint?
8
votes
4answers
307 views
Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$
Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$
I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
1
vote
0answers
43 views
Convergence of an integral involving the fractional part function
It's not too hard to show that $\displaystyle \int_{\textbf{1}}^{\infty} \frac{\text{frac}(x)}{x^{s+1}} \ dx = \frac{1}{s-1} - \frac{\zeta(s)}{s}$ for $s >1$.
And if you take the limit of both ...
0
votes
0answers
35 views
The minimum of a function
Could anyone possibly give me any help with finding the minimum of this function? I believe the result to be $2\pi |n|$ from page 619 of this paper by W. G. C. Boyd.
\begin{equation}
...
1
vote
1answer
78 views
How does one calculate the amount of time required for computation?
For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by
$
...
3
votes
2answers
73 views
Riemann Zeta Function By Hand
This may be a stupid question but is there a way to calculate Riemann's Zeta Function by hand exactly or can you only estimate it?
3
votes
1answer
123 views
The meaning of the Euler Formula for Zeta
Does anybody know about a "meaning" behind the Euler Formula, what does it really say about the primes?
I know that it is in equation to the zeta function and also how it is derived, but cannot find ...
2
votes
1answer
66 views
1+2+3+4+… = -1/12 [duplicate]
Consider the zeta function $\zeta(s)= \sum \limits_{n=1}^{\infty} \frac{1}{n^s}$.
It is established that $ \zeta(-1) = -\frac{1}{12}$.
Reference (Equation 90)
Then we have $ \zeta(-1) = \sum ...
1
vote
0answers
30 views
The sum of the reciprocals of fourth powers [duplicate]
This problem is an extension of the well known basel problem and involves finding the sum of
1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity
Euler managed to prove ...
3
votes
1answer
159 views
Discontinuities of $\sum \frac{x^{\rho}}{\rho}$
H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
5
votes
3answers
97 views
Two trivial questions about zeta function
I have two questions concerning the Riemann zeta function which are rather trivial so if anyone can give me the answers that would be nice, here is what I`m interested in:
1) In the equality ...
15
votes
1answer
186 views
References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$
While extending my calculation techniques, with aid of Mathematica, I found that
\begin{align*}
\int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx
&= -6 \zeta '(-1) ...
1
vote
2answers
112 views
Series expansion of $\zeta(s)$ using the derivatives of $\zeta(0)$
When playing with series expansions, I stumbled upon the following relation:
$$\zeta(s) := \sum \limits_{n=0}^{\infty} \left( \zeta^{(n)}(0) \frac{s^n}{\Gamma(n+1)} \right)$$
that seems to hold for ...
3
votes
1answer
89 views
Find the value of: $ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(x)\,dx $
I'm trying to find to value of:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(x)\,dx
$$
Here's what I've done:
$$
...
9
votes
2answers
195 views
How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?
How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc.
Thank you,
0
votes
0answers
30 views
summation formula involving mertens function
from the residue theorem
$$ M(x) = \sum_\rho \frac{x^\rho}{\rho \zeta'(\rho)} - 2+\sum_{n=1}^\infty \frac{ (-1)^{n-1} (2\pi )^{2n}}{(2n)! n \zeta(2n+1)x^{2n}} $$
asssuming there are no multiple ...
11
votes
2answers
236 views
Can the Basel problem be solved by Leibniz today?
It is well known that Leibniz derived the series
$$\begin{align}
\frac{\pi}{4}&=\sum_{i=0}^\infty \frac{(-1)^i}{2i+1},\tag{1}
\end{align}$$
but apparently he did not prove that
$$\begin{align}
...
1
vote
0answers
67 views
Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?
For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
6
votes
1answer
191 views
how to understand $\log\zeta(s)$ (Riemann zeta function)?
I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region.
Let's assume $Re(s)>1$.
Is $\zeta(s)$ ...
3
votes
1answer
78 views
Calculating an almost Gamma integral
How would you proof that
$$I:=\int_{0}^{\infty}\frac{z^{x-1}}{e^{z}+1}dz=\left(1-2^{1-x}\right)\Gamma(x)\zeta(x)$$
I can rewrite the integral as
...
11
votes
2answers
242 views
A series involves harmonic number
How do we get a closed form for
$$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
7
votes
1answer
90 views
Serre's proof that zeta function is meromorphic
I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as
\begin{align*}
...
2
votes
1answer
77 views
About Euler's formula for Apery number
Euler's formula.
$$\zeta(3)=\frac{\pi^2}{7}\left(1-4\sum_{m\ge 1}\frac{\zeta(2m)}{(2m+1)(2m+2)2^{2m}}\right)$$
I saw this formula in Wikipedia a few months ago. I have searched about Euler's ...
6
votes
2answers
220 views
A tough series: $\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$, need help
I was doing a integral which ends up with a tough series part:
$$\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$$
Mathematica says $$\frac12$$
Which agrees with the anwer...Anyone know how to ...
3
votes
1answer
79 views
How to evaluate $\xi(0)$?
How do I evaluate $\xi(0)$ for the Riemann xi function?
I know $\xi(0) = \xi(1)$ and
$\xi(0) = \tfrac{1}{2} \cdot 0 \cdot (-1) \cdot \Gamma(0) \cdot \zeta(0)$
$\xi(1) = \tfrac{1}{2} \cdot 1 \cdot 0 ...
4
votes
2answers
121 views
Another improper integral
Show that :
$$\int_0^1\frac{(\sin ^{-1}x)^2}{x}\text{d}x=\frac{\pi ^2\ln 2}{4}-\frac78\zeta(3)$$
This integral is in "irresistible integrals" on page 122. I can't prove this one.
7
votes
2answers
213 views
Intuitive explanation with rigorous details why zeta has infinitely many zeros?
I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is:
Simplest possible (least "magic") argument that explains why zeta has ...
5
votes
2answers
206 views
Why does zeta have infinitely many zeros in the critical strip?
I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip.
The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...
2
votes
0answers
68 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}) ...
