Tagged Questions
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?
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 $$ {d^a ...
11
votes
4answers
211 views
Alternating sum of multiple zetas equals always 1?
This is more in the category of "recreational math"...
I was playing with multiple zetas, in the notation of $\zeta(k),\zeta(k,k),\zeta(k,k,k),\ldots$ as given in wikipedia.
Looking at the alternating ...
9
votes
1answer
87 views
A Tough Series $\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$
I have done series with $\zeta(2k)$ and $\zeta(k)$, but I have no idea with this one:
$$\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}.$$
I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
1
vote
0answers
30 views
The sum of the reciprocals of fourth powers [duplicate]
This problem is an extension of the well known basel problem and involves finding the sum of
1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity
Euler managed to prove ...
3
votes
1answer
159 views
Discontinuities of $\sum \frac{x^{\rho}}{\rho}$
H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
11
votes
2answers
236 views
Can the Basel problem be solved by Leibniz today?
It is well known that Leibniz derived the series
$$\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}
...
11
votes
2answers
242 views
A series involves harmonic number
How do we get a closed form for
$$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
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 ...
4
votes
2answers
122 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.
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?
$$
\sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
2
votes
0answers
52 views
Theta series and Riemann Hypothesis
in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf
section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series
$$ F(\phi(x))= ...
1
vote
0answers
100 views
Proof of Euler's general formula for a sum involving harmonic numbers
I have seen this formula, but how to prove this?
$$2\sum\limits_{k=1}^\infty \frac{H_k}{\left( k+1 \right)^m} =m\zeta \left( m+1 \right)-\sum\limits_{k=1}^{m-2}{\zeta \left( m-k \right)\zeta \left( ...
14
votes
1answer
436 views
A nice log trig integral
Show that :
$$\int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\cos x{{\ln }^{2}}\sin x}{\cos x\sin x}}\text{d}x=\frac{1}{4}\left( 2\zeta \left( 5 \right)-\zeta \left( 2 \right)\zeta \left( 3 \right) ...
10
votes
1answer
228 views
Closed form for $\sum_{n=2}^\infty \frac{1}{n^2\log n}$
I had attempted to evaluate
$$\int_2^\infty (\zeta(x)-1)\, dx \approx 0.605521788882$$
Upon writing out the zeta function as a sum, I got
$$\int_2^\infty ...
5
votes
1answer
120 views
Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind
Consider the integral
$$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$
For $n=3$ we have
...
3
votes
4answers
306 views
Formula for partial sum of Riemann zeta function [duplicate]
Possible Duplicate:
Finite Sum of Power?
Suppose $f(s,k) = \sum_{n=1}^k n^{-s}$ is the Riemann zeta function truncated at the k-th term. I read on mathoverflow that there is a formula for ...
4
votes
2answers
92 views
Riemann zeta sums and harmonic numbers
Given the nth harmonic number of order s,
$$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$
It can be empirically observed that, for $s > 2$, then,
$$\sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = ...
7
votes
1answer
217 views
A new formula for Apery's constant and other zeta(s)?
I recently found these Plouffe-like formulas using Mathematica's LatticeReduce. Has anybody seen/can prove these are indeed true?
$$\begin{aligned}\frac{3}{2}\,\zeta(3) &= ...
20
votes
2answers
1k views
Proving an “amazing” claim regarding $\zeta( 3)$ and Apéry's proof
I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$
$$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$
and thus (probably) ...
13
votes
3answers
710 views
Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.
Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$
Computing the Fourier series of $f$ and ...
4
votes
1answer
150 views
Zeta function identity
How does one prove the zeta function identity
$$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
2
votes
2answers
228 views
zeta of three, question about closed form
If $\sum\limits_{n=2}^\infty \frac1{(n^2-n)^3}=10-\pi^2$, then what is the limit in closed form of $\sum\limits_{n=1}^\infty \frac1{n^3}$?
5
votes
2answers
378 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
7
votes
2answers
330 views
How to find $\zeta(0)=\frac{-1}{2}$ by definition?
I would like to know how we can find the following result:
$\zeta(0)=\frac{-1}{2}$
Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$
to find this?
10
votes
2answers
309 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
8
votes
1answer
214 views
An infinite series involving the Zeta Function
I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants:
$$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$
and
...
10
votes
2answers
344 views
Do these series converge to logarithms?
It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$
If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$
Starting:
...
4
votes
0answers
283 views
Is this a relation between the Riemann zeta function and the Prime zeta function?
I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function?
$$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$
...
6
votes
1answer
185 views
What is known about the pattern for $\zeta(2n+1)$?
Related to the question Does $\zeta(3)$ have a connection with $\pi$?:
It is well known that
$$\zeta(2n) = f(2n) \pi^{2n}$$
where $f(n)$ is an function in rationals: (the denominator = OEIS ...
28
votes
5answers
2k views
Does $\zeta(3)$ have a connection with $\pi$?
The problem
Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)?
Details
Several ...
19
votes
2answers
424 views
A double series yielding Riemann's $\zeta$
Can you give me some hints to prove equality:
$$\sum_{m,n=1}^{\infty} \frac1{(m^2+n^2)^2} =\zeta (2)\ G-\zeta(4)=\frac{\pi^2}{6}\ G-\frac{\pi^4}{90}$$
where $\zeta (t):= \sum\limits_{n=1}^{+\infty} ...
25
votes
4answers
3k views
Nice proofs of $\zeta(4) = \pi^4/90$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
