The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
0answers
20 views

Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series? I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...
16
votes
1answer
500 views

Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
-4
votes
0answers
93 views

sequences and series [closed]

I think it is interesting, if we have the formula $$\frac{n (n + 1) (2 n + 1)}{6} = 1^2 + 2^2 + \cdots + n^2 .$$ If the difference between the closest numbers is smaller (let's call is a) we ...
2
votes
0answers
61 views

Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...
5
votes
0answers
106 views

How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question. Take the following, nicely symmetrical, telescoping series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum ...
5
votes
3answers
236 views

Sets of natural numbers with finite intersections and divergent sums of reciprocals

Does there exist an uncountable collection $\Lambda$ of infinite subsets of the set of natural numbers such that (i) any two distinct subsets in the collection have a finite intersection and (ii) the ...
0
votes
0answers
16 views

Sorting circles sizes with points in integers lattice [migrated]

I wrote a program to find all the circles with at least three points located at the integers grid. I started with smallest boxes (width >= height) calculating the radius of a circle with points ...
0
votes
0answers
76 views

An infinite sum which approaches a geometric series

This is a cross-post from http://math.stackexchange.com/questions/1174998/estimating-an-unusual-infinite-sum, which didn't get any useful answers I encountered the following unusual type of infinite ...
3
votes
0answers
101 views

Euler series with milder divergence

Theorema 19 in Euler's memoir "Variae observationes circa series infinitas" says The sum of the reciprocals of the prime numbers is infinitely great but is infinitely times less than the sum of the ...
11
votes
2answers
451 views

What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...
3
votes
1answer
217 views

New series for $1/\pi$ based on Ramanujan's ideas

In his classic paper "Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in ...
5
votes
1answer
175 views

Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function $$ \begin{align} \displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}, \quad \Re a ...
2
votes
1answer
69 views

Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following ...
3
votes
0answers
213 views

Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...
1
vote
1answer
76 views

Series estimate

Let $\theta\in(0,1)$ be given. I define for $a>0$ and $\lambda \ge 1$, $ S(\lambda,a )=\sum_{k\ge 1} k^{\frac12-\theta}e^{-a\vert k-\lambda\vert}. $ I want to prove that $$ ...
10
votes
2answers
225 views

Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.

Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that ...
7
votes
0answers
280 views

About the first decimal of $\sqrt {n!}$

Do we have : $$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$ Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
35
votes
2answers
1k views

Why does this sequence converges to $\pi$?

One of my daughters was having a small programming exercise. Let's consider following algorithm: Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$. Remove every $2$nd number. From the ...
1
vote
0answers
45 views

closed form for a series with binomials and primes

does the series $\sum_{n=0}^\infty p^n \binom{x}{p^n}$ have a closed form ? ($p$ prime) this is a special case of $\sum_{n=0}^\infty p^n \left(\sum_{k=p^n}^{p^{n+1}-1}a_k\binom{x}{k}\right)$ with the ...
5
votes
0answers
115 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
7
votes
2answers
447 views

Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
4
votes
0answers
187 views

Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas: 1) The ...
10
votes
1answer
616 views

Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$. We denote the least integer not covered by the arithmetic ...
1
vote
1answer
176 views

Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
2
votes
5answers
348 views

Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
0
votes
0answers
107 views

this sequence $A_{n}$ have recursive relations?

Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form ...
12
votes
4answers
654 views

How to calculate the infinite sum of this double series?

I'm calculating this double sum: $$ \sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2} $$ I know the answer is $$ \frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16} $$ which can be ...
28
votes
3answers
1k views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit, ...
22
votes
7answers
2k views

What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
3
votes
1answer
267 views

Does this function have any exponential growth?

Has anyone seen any function of the following type? $$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$ The question is whether for some constant ...
5
votes
1answer
286 views

The closed form of $\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$

The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$ where $\psi(n)$ is digamma function arose in the evaluation of an integral I posted on MSE, ...
3
votes
1answer
278 views

A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm

Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...
6
votes
3answers
397 views

Asymptotic formulas for Monster-related modular functions?

Define the following, $$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$ $$j_{2A}(\tau) ...
2
votes
2answers
230 views

Is this infinite series related to some well-known special functions?

Please allow me to resort once again to the expertise of the MathOverflow community : During research I encoutered the following infinite series : $$\sum_{n=-\infty}^{+\infty} ...
2
votes
0answers
195 views

Minimizing $\{0,1\}$-sequence permutations

Explanation: For a given bit sequence $f$, reposition the bits as to minimize $G$ which can be thought of as a measure of how poorly proportional $f$ is to each of its subsequences. Let $p \in ...
0
votes
0answers
46 views

bounds on a series with binomial coefficients

I have the following series $\sum\limits_{l=1}^n {n\choose l} \alpha^{\beta^l}$ where $\alpha > 0$and $0 \leq \beta \leq 1$. Can anybody guide me how I can evaluate it or find some tight upper ...
4
votes
2answers
242 views

What is known about this series?

I recently came across the following function which intrigues me: \begin{equation} f(\alpha):=\sum_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}. \end{equation} For $-1\leq \alpha\leq 1$ this function is ...
5
votes
2answers
229 views

expression for infinite series with powers of factorial in denominator

The series $$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$ has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
2
votes
2answers
107 views

Finite series with reciprocal factorials

I asked the question at MSE http://math.stackexchange.com/questions/982388/simple-finite-series-with-reciprocal-factorials but got no answer or comment (it is not a homework). I'm trying to find the ...
2
votes
3answers
391 views

How did the summation operation come into use? [closed]

So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...
2
votes
0answers
105 views

The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...
0
votes
2answers
93 views

Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...
1
vote
1answer
185 views

Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$. Let $k\ge0$ be a nonnegative integer. If we add another factorial ...
7
votes
1answer
242 views

Is this series well known?

I recently encountered the following function $$ f(t) = \sum_{n=0}^\infty \frac{t^{n^2}}{n^2!}. $$ It seems familiar, though I cannot remember where I might have seen it before. I would like to know ...
1
vote
2answers
68 views

Summation mollifier to ensure a certain alternating series has the correct value

I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions: (1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$ (2) $f(n,M) ...
0
votes
0answers
55 views

$\eta(s)$ expressed as an 'alternating' sum of Hurwitz Zetas. Why does it only work for sums with an even number of terms?

It is known that: $$\zeta(s)= a^{-s}\,\sum_{k=1}^{a} \zeta_H\left(s,\frac{k}{a}\right)$$ is valid for all $a \in \mathbb{N}$ and all $s \in \mathbb{C}\,/1$, with $\zeta_H$ being the Hurwitz zeta ...
3
votes
2answers
344 views

Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function. Numerical evidence suggests these identities: $$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...
1
vote
0answers
54 views

Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here. In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London ...
0
votes
0answers
68 views

Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem: Is there a method to extend the notion ...
8
votes
0answers
173 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...