All Questions
24
questions
-1
votes
1
answer
70
views
How to give this sum a bound?
Let $x,y\in\mathbb{Z},$ consider the sum below
$$\sum_{x,y\in\mathbb{Z}\\ x\neq y}\frac{1}{|x-y|^{4}}\Bigg|\frac{1}{|x|^{4}}-\frac{1}{|y|^{4}}\Bigg|,$$
is there anything I could do to give this sum a ...
- 329
2
votes
0
answers
97
views
Fractional part of a sum
Define for $n\in\mathbb{N}$ $$S_n=\sum_{r=0}^{n}\binom{n}{r}^2\left(\sum_{k=1}^{n+r}\frac{1}{k^5}\right)$$
I need to find $\{S_n\}$ for $n$ large where $\{x\}$ denotes the fractional part of $x$.
$$...
- 910
0
votes
2
answers
62
views
Comparing integral with a sum
Show that \begin{equation}\sum_{m=1}^k\frac{1}{m}>\log k.\end{equation}
My intuition here is that the LHS looks a lot like $\int_1^k\frac{1}{x}\textrm{d}x$, and this evaluates to $\log k$. To ...
- 135
2
votes
0
answers
63
views
Cases where transcendental numbers can add up to a rational number? [closed]
Other than sums like $π + (1 - π)$, obviously. Can two transcendental numbers add up to a rational number? Or how about an infinite series of them?
- 453
6
votes
2
answers
220
views
Finding a closed form for $\sum_{k=1}^\infty\sum_{n=k}^\infty\left(\frac{(-1)^k}{k^3\binom{n+k}{k}\binom{n}{k}}(\frac1{n^2}-\frac1{(n+1)^2})\right)$
Consider the sum $$\sum_{n=k}^{N} \frac{1}{\binom{n+k}{k}\binom{n}{k}}\left(\frac{1}{n^2}-\frac{1}{(n+1)^2}\right) $$
Using the above or otherwise I need a closed form for $$\sum_{k=1}^{\infty}\sum_{n=...
- 910
1
vote
0
answers
82
views
Farey Sequence and Mertens function
Mertens function $M(n)$ is defined as the cumulative sum of Möbius functions $\mu(k)$:
$$M(n)=\sum_{k=1}^n\mu(k)$$
and is profoundly related to the Riemann hypothesis. A nice alternative formula (...
- 827
1
vote
2
answers
83
views
Find all the integers which are of form $\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}, a,b,c\in \mathbb{N}$, any two of $a,b,c$ are relatively prime.
I have a question which askes to find all the integers which can be
expressed as
$\displaystyle \tag*{} \dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}$
where $a,b,c\in \mathbb{N} $ and any two of $a,b,...
- 921
4
votes
0
answers
97
views
Convergence rate to the average value for a sequence that is uniformly distributed modulo $1$
Let $x_n$ be a sequence of real numbers that asymptotically follow a uniform distribution modulo $1$ (for example, $x_n=n\sqrt{2}$ with $n$ positive integer $\leq N$ and $N\rightarrow \infty$). It is ...
- 17.1k
0
votes
2
answers
244
views
How is $ B_n = 1- \sum_{k=0}^{n-1} \binom{n}{k} \frac{B_k}{n-k+1} $ Where $B_n$ are the Bernoulli Numbers with $B_1 = \frac{1}{2}$
So I was browing Wikipedia just looking at Bernoulli Number identities and I stumbled across this
$$ B_m^+ = 1- \sum_{k=0}^{m-1} \binom{m}{k} \frac{B_k^+}{m-k+1} $$
The Wikipedia page said that this ...
0
votes
1
answer
40
views
Need help in proving an inequality related to logarithms
I am self studying analytic number theory from class notes of a senior and in it I am unable to deduce an inequality which is not proved .
Assume $b(m) = \sum_{s, t} \frac{1} {log(s) log(t) } $...
user775699
2
votes
0
answers
58
views
Prove the following summation identity
Let $$\sigma_k (n)= \sum_{j = 1}^{n} j^k$$ and $$\Psi_k (n)= \sum_{j = 1}^{n} \sigma_k (j).$$
Prove the following without use of the respective polynomial equations associated with $\sigma_k$ and $\...
- 21
6
votes
1
answer
242
views
Part 1: Does the arithmetic mean of sides right triangles to the mean of their hypotenuse converge?
Let $a_k<b_k<c_k$ be the $k$-th primitive Pythagorean triplet in ascending order of the hypotenuse $c_k$. Define
$$
l = \frac{b_1 + b_2 + b_3 + \cdots + b_k}{c_1 + c_2 + c_3 + \cdots + c_k}, \...
- 20.8k
5
votes
0
answers
157
views
What is the sum of the reciprocal of the hypotenuse of Pythagorean triangles?
A primitive Pythagorean triplet is a triplet $a^2 + b^2 = c^2$ be where $a,b,c$ have no common factors and is generated by $a = x^2 - y^2, b = 2xy, c = x^2 + y^2$ where $x > y, \gcd(x,y) = 1$. My ...
- 20.8k
15
votes
1
answer
470
views
Sum of the inverse squares of the hypotenuse of Pythagorean triangles
What is the sum of the series
$$ S = \frac{1}{5^2} + \frac{1}{13^2} + \frac{1}{17^2} +
\frac{1}{25^2} + \frac{1}{29^2} + \frac{1}{37^2} + \cdots $$
where the sum is taken over all hypotenuse of ...
- 20.8k
19
votes
1
answer
1k
views
What is the closed form of this sum?
What is the closed form of this sum?
$$
S = \sum_{k\ge1, r>s\ge 1}\frac{1}{k^2(r^2 + s^2)^2}
$$
Note: Though originally posted for Pythagorean triplets, their was an flaw in the question which ...
- 20.8k