All Questions
Tagged with analytic-number-theory sequences-and-series
88 questions
4
votes
1
answer
213
views
Asymptotic behavior of weighted sums involving the fractional part function
Currently, I am studying the asymptotic behavior of sums of the form
\begin{equation}\label{eq1}\tag{1}
\sum_{k=1}^{n-1} f(n-k) \left\{ \frac{n}{k} \right\}
\end{equation}
In this context, based on ...
- 611
2
votes
0
answers
121
views
Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum ...
- 4,829
7
votes
4
answers
793
views
Must bounded sequences be well-distributed to most *composite* moduli?
Let $\{a_n\}_{n=1}^N$, $|a_n|\leq 1$. Let $Q=\sqrt{N}$. Then $a_n$ is well-distributed modulo most prime $p\leq Q$, in the following sense:
$$\sum_{p\leq Q} \frac{1}{p} \left(\frac{1}{N/p} \sum_{\...
- 20.2k
2
votes
0
answers
157
views
What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
- 4,829
1
vote
1
answer
101
views
Finding $\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma$ from the generating function of $\zeta(•)$
In equation (130) of this page, the identity $$\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma \label{1} \tag{1} $$ is stated. Here, $\zeta(\cdot)$ is ...
- 4,829
2
votes
1
answer
161
views
Closed form expression for this zeta-like series involving GCD and LCM
I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$:
$$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
- 634
3
votes
2
answers
511
views
Exotic series for some mathematical constants from String Theory
Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and ...
- 2,836
7
votes
1
answer
332
views
A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?
This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function:
$$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\...
- 4,169
3
votes
4
answers
497
views
Asymptotic for Ramanujan's $\tau$-function
The Ramanujan's $\tau$-function is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$.
Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
- 317
2
votes
0
answers
131
views
Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,098
0
votes
1
answer
259
views
Finding a strictly increasing Collatz sequence of arbitrary length [closed]
Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I ...
2
votes
1
answer
197
views
2D lattice sum with numerator
I've been struggling a bit with a double sum that arose as the trace of an operator:
$$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$
where $n$ is an even natural number. Is there ...
- 31
2
votes
1
answer
152
views
The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$
Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...
- 852
1
vote
0
answers
63
views
Set from a diophantine equation with similar statistics to primes
While doing some computational calculations with some diophantine equations, I came across with some sequences from solutions of quartic and quintic equations with slowly decreasing frequency, similar ...
3
votes
1
answer
474
views
Curious infinite product, convergence, connection to prime numbers
I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with ...
- 3,098
6
votes
1
answer
494
views
(Explicit) Tauberian theorems: removing $(\log x/n)$
Say that $\{a_n\}_{n\geq 1}$, $|a_n|\leq 1$, are such that $$\left|\sum_{n\leq x} a_n \log \frac{x}{n}\right|\leq \epsilon x\quad\text{for all $x\geq x_0$.}$$ What sort of bound can we deduce on $S(x)=...
- 20.2k
1
vote
1
answer
96
views
On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function
In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
2
votes
1
answer
290
views
One series converges iff the other converges
In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges
$$
\begin{split}
\sum_{1<n\leq N}\frac{a_{n}}{\...
- 31
1
vote
0
answers
187
views
$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$
It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that
$$\lim_{x\to \infty} \sum_{n\leq x} \frac{(\log n)^k}{n} = \int_1^x \frac{(\log t)^k}{t} + C + O\left(\frac{(\log x)^k}{...
- 20.2k
4
votes
1
answer
219
views
Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$
Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
- 5,470
3
votes
3
answers
483
views
Growth of the coefficients of the inversion of the $j$-invariant function
We have the $j$-invariant defined as
I have that
$$
j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k,
$$
where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.
The inversion ...
- 515
5
votes
1
answer
435
views
Limit on a certain double sum
While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense). Notice first this well-known fact in context
$$\sum_{n,m\geq1}\frac1{...
- 43.2k
1
vote
0
answers
91
views
Any known relations to this doubly exponential constant?
Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:
$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...
- 2,689
3
votes
0
answers
219
views
On partial sums of the Ramanujan sums
Let $n$ be a positive integer and $c_{m}(n)$ denote the $m^{th}$ Ramanujan sum at $n$. What is the best known estimate for $\sum_{m=1}^{N} c_{m}(n)$?
- 31
2
votes
0
answers
448
views
Conjecture: The sequence {$π(2n+1)!$} is equidistributed in the interval (0,1)
Let $n\in\mathbb{N}$.
From the book "Uniform Distribution of Sequences" (available here) by L. Kuipers and H. Niederreiter, (from pg. 8) I found that for any irrational $\theta$, the ...
1
vote
1
answer
387
views
Under a condition, $\frac{1}{b } = \sum_{n=1}^{\infty}\frac{1}{a_{n}}$ will never happen
Conjecture:
There is no $b,\{a_n\}_{n=1}^{\infty}$ such
that $b,a_n \in \mathbb{N}^+, a_{n+1}\ge a_n$,
$$\lim_{n\rightarrow \infty}\frac{a_{n+1}}{a_{n}}=\infty\qquad\text{and}\qquad\frac{1}{b}= \...
- 129
3
votes
1
answer
134
views
Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$?
Is it possible to find an estimate of the summation
$$s(n)=\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$$
where $\varphi(n)$ is the totient function and $p_k$ the k-th prime?
The corresponding series seems ...
- 1,008
11
votes
2
answers
726
views
What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?
Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges?
Can one compute a few digits based on euristic considerations or plausible ...
- 5,103
2
votes
0
answers
154
views
How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?
It has been discovered long ago that
$$\sum_{n=2}^{\infty} \big(\zeta(n) - 1\big) = 1. \label{1} \tag{1} $$ More recently, a generalization of this result with binomial coefficients has been obtained: ...
- 4,829
1
vote
1
answer
2k
views
About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$
The following real-valued functions are closely related to the zeros of $\zeta(s)$ in the critical strip $\frac{1}{2}<\Re(s) < 1$.
$$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\...
- 3,098
2
votes
0
answers
162
views
Limit of infinite power tower $\lim_{n \rightarrow +\infty}\frac{a_0^{a_1^{a_2^{^{.^{.^{a_{n}}}}}}}}{b_0^{b_1^{b_2^{^{.^{.^{b_{n}}}}}}}}$
Let $\{a_n\}_{n \in \mathbb{N}}$ be a sequence of natural numbers. Let us define a function which roughly "make a tower of powers out of $\{a_n\}_{n \in \mathbb{N}}$", i.e.
$$F:\mathbb{N}^{\...
- 1,343
1
vote
0
answers
78
views
Concerning the coefficient $[q^n]\sum_{n\ge1}\frac{(aq)^n}{(1-bq^n)^2}$
I posted this on MSE, and not even @ParamanandSingh could answer, so I thought I should post it here
Background:
While trying to answer this question, I came up with a question of my own.
Let $|a|,|b|,...
- 283
2
votes
0
answers
94
views
Rational zeta series and differential-difference equations
In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by ...
- 4,829
2
votes
0
answers
422
views
Sequences with high densities of primes: how to boost them to get even more and larger primes
I propose a methodology to help find large prime numbers with a much higher probability than picking up random numbers and testing them for primality. This would help speed up prime number generators ...
- 3,098
3
votes
1
answer
330
views
A question on an identity relating certain sums of Harmonic numbers
In the description of this question, it was established that \begin{align} \sum_{n=2}^{\infty} (\zeta(n)^{2}-1) &=
\frac{7}{4} - \zeta(2) + 2 \sum_{m=1}^{\infty} \frac{H_{m-1- \frac{1}{m}} - H_{-...
- 4,829
10
votes
1
answer
731
views
What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?
A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$
Many more identities can be found in articles by e.g. Borwein and Adamchik &...
- 4,829
24
votes
1
answer
2k
views
Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 429
7
votes
0
answers
332
views
Do generalizations of the identity $\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) = 1 $ exist?
On p. 263 of Borwein's paper entitled “Computational Strategies for the Riemann zeta function”, the following identity is stated: $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1 . \qquad \...
- 4,829
5
votes
0
answers
343
views
Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
2
votes
2
answers
385
views
What is the growth rate of the sum of powers of distinct primes closest to a given a integer?
Let $n$ be a positive integer, and
$$2 = p_1 < p_2 < \dots < p_m \le n$$
be the sequence of all primes less than or equal to $n$.
For each index $j$ let $p_j^{e_j}$ be the largest power of $...
- 557
11
votes
1
answer
497
views
Sign of the function $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$
It is well-known that the Mertens function $M(n)=\sum_{k=1}^n\mu(k)$ changes sign infinitely many times when $n\rightarrow +\infty$. Let $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$, then $\lim\limits_{n\...
- 622
5
votes
1
answer
491
views
Is the parity of $\omega(n)$ equally distributed?
I recently learned that the prime omega function $\Omega(n)=\Omega\left(p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}\right)=\alpha_1+\alpha_2...+\alpha_k$ is very well studied. In particular, we know ...
- 2,902
4
votes
0
answers
924
views
What fraction of fractions does Cantor's famous sequence enumerate?
Cantor's famous sequence
$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$
provides a ...
user112109
11
votes
3
answers
866
views
Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$
I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
- 2,902
5
votes
3
answers
1k
views
Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?
Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
- 1,126
4
votes
1
answer
247
views
Are there any extensive treatments on rational zeta series?
I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
- 4,829
2
votes
1
answer
142
views
What is the collection of series that amount to $\gamma$ deduced by Ramanujan?
On p. 20 of an article by Borwein et al., it is stated that Ramanujan could generalize the following formula due to Glaisher $$\gamma = 2 - 2\log2 -2\sum_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$...
- 4,829
-2
votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
0
votes
0
answers
85
views
Mean and logarithmic values for arithmetic function
Define the mean value for function $f$ as $\lim \limits_{x \to \infty} \frac{1}{x} \sum \limits_{n \leq x} f(n)$ if the limit exists denoted as $M_f$
Define the logarithmic value for function $f$ as $...
- 1
0
votes
1
answer
335
views
On partial sum estimate on the series $S(p,q;s)=\sum_1^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$ and other Generalizations
Consider the following sum :
$$S(p,q;s)=\sum_{n=1}^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$
Here , $p$ is a variable w.r.t which we are going to analyse the sum.
$s$ is another parameter with ...
- 375