All Questions
Tagged with nt.number-theory real-analysis
31 questions
46
votes
7
answers
10k
views
Are some numbers more irrational than others?
Some irrational numbers are transcendental, which makes them in some sense "more irrational" than algebraic numbers. There are also numbers, such as the golden ratio $\varphi$, which are poorly ...
- 1,823
122
votes
5
answers
27k
views
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 4,535
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
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 5,470
11
votes
1
answer
1k
views
Extending an assignment property from Q to R (or C)
Property of any odd number of nonnegative integers:
Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
- 7,831
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
239
votes
14
answers
76k
views
Have any long-suspected irrational numbers turned out to be rational?
The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration ...
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
44
votes
3
answers
4k
views
Smooth functions for which $f(x)$ is rational if and only if $x$ is rational
A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$, ($f \in C^\infty$), such that $f$ maps rationals to rationals and ...
- 543
29
votes
1
answer
1k
views
About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$
I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far.
I ...
- 483
49
votes
3
answers
6k
views
The Hardy Z-function and failure of the Riemann hypothesis
David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...
- 13.1k
41
votes
6
answers
9k
views
"Long-standing conjectures in analysis ... often turn out to be false"
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
34
votes
1
answer
2k
views
Ruling out the existence of a strange polynomial
Does there exist a polynomial $f \in \mathbb{Z}[x,y]$ such that
$$\displaystyle f(a,b) > 0 \text{ for all } a,b \in \mathbb{Z}$$
and
$$\displaystyle \liminf_{(x,y) \in \mathbb{R}^2} f(x,y) = -\...
- 26.9k
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 19.6k
28
votes
3
answers
2k
views
Does $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converge to zero when $\alpha$ is irrational?
I came across a problem concerning about the convergence of products. I wonder if the complex series $a_n=\prod^n_{k=1}(1-e^{k\alpha \pi i})$ converges to zero when $\alpha$ is irrational. Of course, ...
- 1,565
22
votes
2
answers
2k
views
Is a real power series that maps rationals to rationals defined by a rational function?
Suppose that the function $p(x)$ is defined on an open subset $U$ of $\mathbb{R}$ by a power series with real coefficients. Suppose, further, that $p$ maps rationals to rationals. Must $p$ be defined ...
- 6,199
15
votes
4
answers
1k
views
Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
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
23
votes
1
answer
3k
views
Does the average primeness of natural numbers tend to zero?
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value ...
- 5,470
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
11
votes
1
answer
436
views
How many numbers $\le x$ can be factorized into three numbers which form the sides of a triangle?
Note: Posting in MO since it was unanswered in MSE
Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \...
- 5,470
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
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
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
8
votes
1
answer
783
views
Real-rooted polynomials
I proposed this question at MO which was resolved neatly by Gerald Edgar in the form
$$
u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k).$$
Now that we ...
- 43.2k
8
votes
3
answers
747
views
How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
5
votes
1
answer
425
views
"Interlacing property" of certain polynomials
I posted this question on MO which was quickly and decidedly answered by Noam D. Elkies.
Once more referring to the same set of polynomials
$$u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\...
- 43.2k
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
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
3
votes
1
answer
251
views
Congruence modulo 2 for q-series
This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks.
I would like to ask:
QUESTION. Is this congruence true ...
- 43.2k
3
votes
0
answers
232
views
When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
- 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
0
votes
0
answers
45
views
A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303