All Questions
Tagged with nt.number-theory real-analysis
208 questions
7
votes
0
answers
221
views
integrality of a Riccati-type equation
The following is a problem we were unable to prove and left stated in the paper
"Arithmetical properties of a sequence arising from an arctangent sum", J. Numb. Theory 128 (2008) 1807–1846.
Define ...
- 43.2k
3
votes
2
answers
491
views
Unknown bias in a distribution related to prime numbers
If $n$ is composite then $\phi(n) < n-1$, hence there is at least one divisor $d$ of $n-1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trvially, if $n$ is prime then ...
- 5,470
3
votes
1
answer
290
views
Fluctuating constants
Let $p_k$ be the $k$-th prime number, $\gamma$ be the Euler-Mascheroni constant and $M$ be the Meissel–Mertens and let $m$ be the integer part of $\log p_n$. We can show that
$$
\sum_{r=1}^{m} \frac{...
- 5,470
46
votes
2
answers
6k
views
Is the following identity true?
Calculation suggests the following identity:
$$
\lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ ...
- 2,183
9
votes
1
answer
299
views
Sequence of nested sets in $[0, 1]$ with bound on gaps
What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...
user78817
2
votes
1
answer
216
views
Ask for a special function related to the error function
I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...
- 1,649
2
votes
1
answer
119
views
On cluster points of a particular sequence
This is the sequel of a previous question.
Let us consider the sequence
$$
\xi_n = 2n \{n\xi\}-n,
$$
where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part.
Do ...
- 459
28
votes
3
answers
2k
views
Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
- 1,565
22
votes
3
answers
2k
views
The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity
What is the origin of the Ramanujan's approximate identity $$\pi^4\approx 2143/22,\;\;\tag 1$$ which is valid with $10^{-9}$ relative accuracy? For comparison, the relative accuracy of the well known $...
- 16.5k
3
votes
1
answer
367
views
Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros
Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum_{...
- 219
5
votes
1
answer
680
views
When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...
- 3,443
14
votes
3
answers
878
views
Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$
Let $a_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a_i\neq a_j$.
I have the following theorem:
For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which $[a_k,...
user79942
4
votes
1
answer
143
views
Mean value of a function associated with continued fractions
Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?
- 3,443
4
votes
1
answer
251
views
Superadditivity of the lower density
Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) $\mu^\star(\...
- 1,511
0
votes
0
answers
344
views
Beurling density $D(X)$ of $X=\{x_j\in\mathbb R, \ |x_j-x_{i}|>\gamma>0: \ i,j\in\mathbb Z\}$
Beurling density of set $X$ is defined (see, for example "Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces" by Aldroubi and Grochenig) as:
$$D(X)=\lim_{r\rightarrow \infty} \inf_{y\in\...
- 297
5
votes
2
answers
957
views
Dirichlet's approximation only using prime power as denominator
I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...
- 1,692
3
votes
1
answer
222
views
Asymptotic for binomial sums
Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use ...
- 33
-1
votes
1
answer
173
views
For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$
Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
- 11
0
votes
1
answer
125
views
Discussion for the sign of a specific sum
Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to ...
- 1,257
8
votes
1
answer
398
views
A question involving e, floor, and all x > 0
Is $\lfloor(x+1/2)e\rfloor = \lfloor(x+1)(1+1/x)^x\rfloor$ for all $x > 0$?
The question occurred in connection with (nonhomogeneous) Beatty sequences, $\lfloor nr+h\rfloor$, where irrational $r&...
- 3,443
9
votes
1
answer
564
views
$L^1$ norm of exponential sum of $n^2 x$
What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.
- 2,207
54
votes
4
answers
3k
views
When has the Borel-Cantelli heuristic been wrong?
The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...
- 11.4k
2
votes
1
answer
223
views
Infinite sum of asymptotic expansions
I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq \dfrac{1}{k^...
- 21
1
vote
1
answer
260
views
Nepero game (by Yacov Perelman)
I have already posted this question time before on stackexchange, but didn't receive a definitive solution.
So this is the game: consider a positive integer number $n$ and divide it in a finite ...
- 11
-1
votes
1
answer
230
views
Prove that $\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2})$ [closed]
Let $a,b\in\mathbb{Z}$, and $f\in C^2([a,b])$ such that $|f''(t)|\asymp \lambda$ for $a\le t\le b$. Prove that
$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}).$$
...
- 31
-1
votes
1
answer
222
views
Does the divergence of the sum of reciprocals of a set of integers imply this density statement about the set?
Suppose $A \subseteq \mathbb{N}$ is such that $\displaystyle{\sum_{n \in A} n^{-1}} = \infty$. Suppose $B \subseteq \mathbb{N}$ is infinite.
Is there a set $X \subseteq [1,\infty)$ and a increasing ...
- 11
2
votes
1
answer
247
views
is $x_{n}\ll \overline{x}_{n}^{2}$?
This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\...
- 6,984
15
votes
4
answers
1k
views
Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...
- 41.1k
2
votes
1
answer
416
views
What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$? [closed]
I have a question: What is the value of $\sum _{n=1}^{\infty \:}\frac{n!}{n^n}$?
Only I've calculated the following identity:
$$\sum _{n=1}^{\infty \:}\frac{n!}{n^n}=\int _0^{1}\left(1+x\cdot \ln \...
- 21
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 113
0
votes
2
answers
145
views
Equivalent of Stirling-like numbers
let $b_{n,k}$ be the numbers defined formally by $$X^n=\sum_{k=0}^n b_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod_{k=0}^{n-1}(X-k)$.
I am looking for an equivalent of $b_{n,k}$ when $k$ ...
- 3,998
7
votes
1
answer
397
views
Fourier expansion of Takagi-function (everywhere non differentiable function).
Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...
- 181
2
votes
0
answers
229
views
Analytic varieties for the primes and the twin primes
I am wondering what real and complex analysis say
about the primes and twin primes.
According to Wikipedia
analytic variety is defined locally as the set of common zeros of finitely many analytic ...
- 25.4k
19
votes
5
answers
1k
views
Floors of powers of reals, how much do the first few determine the next?
Call an integer sequence $\mathbf{x}=\left( x_1,x_2,\cdots \right)$ feasible if it is $f(r)=\left(\lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\ldots, \lfloor r^n \rfloor, \ldots \...
- 30.1k
12
votes
1
answer
991
views
The geometric-mean factorial
Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$,
where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing
$\odot$ with other ...
- 151k
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 19.6k
1
vote
1
answer
603
views
The smallest altitude amongst the triangles formed by points in the unit circle
Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this ...
- 19
0
votes
1
answer
250
views
Equation in integers of irrational degree
Are there any algebraic irrational numbers in $\{log_xy|x,y\in\mathbb{N},x,y\geq2\}$?
- 7,377
8
votes
1
answer
838
views
Density of prime pairs whose gap is less than the average gap
By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
- 5,470
10
votes
2
answers
766
views
When polynomial f(x^2) can be factored as g(x)·g(-x) ?
In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where $...
- 34.3k
2
votes
3
answers
1k
views
on the set of numbers generated by integer linear combination of two real numbers.
Let $b > a > 0$ be two real numbers. I am interested in the set of numbers
$X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$.
What ...
- 29
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 19.6k
11
votes
1
answer
1k
views
Extending an assignment property from Q to R (or C)
Property of any odd number of nonnegative integers:
Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
- 7,831
13
votes
2
answers
2k
views
Asymptotics of the n-th prime using the gamma function
In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that.
$$
p_n = n \...
- 5,470
0
votes
1
answer
939
views
Asymptotic equivalence for functions with zeros
I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.
...
- 2,480
46
votes
4
answers
8k
views
Why could Mertens not prove the prime number theorem?
We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 5,470
1
vote
2
answers
474
views
Chebyshev's Theorem
Hi,
I´m looking for Chebyshev´s theorem which says that the inequality $|x(k)-y|<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\...
- 11
7
votes
1
answer
876
views
A curious definite integral
I was playing around with $\mathcal{I}=\int_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that
$$
\mathcal{I} =
\begin{cases}
\frac{1}{...
- 829
4
votes
0
answers
273
views
Real Analytic Function and nth Prime
It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...
- 153
3
votes
1
answer
500
views
Hausdorff measure on product spaces of p-adic integers
This question came up (unexpectedly) in a problem I was working on a few years ago. It may not be too difficult but I never got around to figuring out the answer, because all I needed at that time was ...
- 1,723