All Questions
Tagged with nt.number-theory real-analysis
208 questions
1
vote
1
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341
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Does normalcy in one base imply normalcy in any other base?
Let $b\geq 2$ be an integer. A real $r \in [0,1]$ is said to be normal with respect to $b$ if every finite string made from the elements $\{0,\ldots,b-1\}$ appears in the $b$-ary expansion of $r$.
Are ...
- 48.9k
2
votes
1
answer
332
views
Convergence of $\sum(n^p\sin^qn)^{-1}$
I've been recently interested in the problem of convergence of the function in such form: $\displaystyle \sum_{n=1}^\infty\frac1{n^p\sin^q n}$.
I saw there's been discussion here when $p=3, q=2$ and $...
- 21
4
votes
1
answer
246
views
Is $C_n$ infinitely log-convex?
A sequence $a_n$ is called log-convex if $\mathcal{L}(a_n):=a_{n+1}a_{n-1}-a_n^2\geq0$ for all $n$; it is infinitely log-convex provided that all the iterates $\mathcal{L}^k(a_n)$ are still log-convex,...
- 43.2k
3
votes
1
answer
702
views
$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
- 3,098
11
votes
1
answer
391
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Is this set dense in [0,+∞)?
We define $A=\{ \frac{c}{rad(abc)}: a, b > 0, c=a+b, gcd(a, b)=1 \}$.
Is the set $A$ dense in $[0, +\infty)$?
Does $\overline{A}$ have interior? Here $\overline{A}$ is the closure of $A$.
A well-...
- 577
29
votes
2
answers
4k
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Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
- 43.2k
2
votes
0
answers
448
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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 ...
3
votes
0
answers
138
views
The mystery of the jumps of functions with the prescribed jumps: Eisenstein series and hidden symmetries(?)
Say that a function $f(t)$ “changes only by jumps” if $f(t) + \text{const} = C ∑_k j_k θ(t-t_k)$ for a certain constant $C$. Here $θ(t)$ is the Heaviside
step function which has a jump 1 at $t=0$ (it ...
- 455
6
votes
0
answers
283
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Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
- 161
7
votes
1
answer
466
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Is there a dense planar rational point set within which the distance of any two points is an irrational number?
i.e. could we find a subset $X\subset \mathbb{Q}^2$ such that $\overline{X}=\mathbb{R}^2$ and that for any $x,y\in X$ the distance $|x-y|$ is an irrational number?
I'm considering the following ...
user178596
10
votes
2
answers
598
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Taylor series with coefficients in $\mathbb{Q}$
Is there a sequence of rational numbers $a_0, a_1, \dotsc$ such that $\sum\limits_{i\geq 0}a_i x^i$ converges absolutely to $2^x$ for every $x\in \mathbb{Z}$?
- 141
9
votes
1
answer
205
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Reaching real numbers from other real numbers by changing a small number of digits in the base b expansion
Given a real number $r$, and an integer $b$>0, we can define $B_b(r)$ as the set of numbers which are obtained from $r$ by writing $r$ in base $b$ and then altering a density zero subset of its ...
- 6,969
3
votes
1
answer
215
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Laurent polynomials: what is the correspondence here?
Given a Laurent polynomial $f$, denote the number of terms by $\#f$ and let $\widehat{CT}(f)$ stand for the value of the constant term in $f$. For example, if $f(x,y)=2-\frac{y}x-\frac{x}y$ then $\#f=...
- 43.2k
2
votes
2
answers
269
views
Ratios of polynomials and derivatives under a certain functional
Let $p(x)$ be a polynomial of degree $n>2$, with roots $x_1,x_2,\dots,x_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping
$$V_m(p)=\sum_{1\leq i<j\...
- 43.2k
1
vote
2
answers
97
views
"Oddity" of a log-Bessel sequence happening at powers of $2$
Define the sequence $b_1=1$ and
$$b_n=\sum_{k=1}^{n-1}\binom{n-1}k\binom{n-1}{k-1}b_kb_{n-k}.$$
By now, there is enough in the literature that $C_n$ is odd iff $n=2^k-1$ for some $k$ where $C_n$ are ...
- 43.2k
4
votes
1
answer
188
views
Is there a generalization of these q-series identities?
Denote $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$.
The below three identities are known.
\begin{align*}
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(q;q)_n}
&=1-\sum_{n\in\mathbb{Z}}(-1)^nq^{\...
- 43.2k
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
$$u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\...
- 43.2k
8
votes
1
answer
783
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Real-rooted polynomials
I proposed this question at MO which was resolved neatly by Gerald Edgar in the form
$$
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).$$
Now that we ...
- 43.2k
4
votes
1
answer
395
views
Mertens formulas aren't enough for prime number theorem
For the primes it's true that
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln x + M + O(1/\ln x)
$$
where, $M$ is suitable constant, and, moreover, the prime number theorem gives that
$$
\lim_{x\to\infty}\frac{\...
- 41
-2
votes
1
answer
180
views
Is there a function $f$ that is a finite sum of functions with finite products of the inputs of $f$ as inputs with this property?
Note: This question aims to be a generalization of Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? and Is it possible to create a polynomial $p(x)$ with this ...
- 265
-1
votes
1
answer
96
views
Limiting points of elementary set
I consider the following set
$$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$
Is it possible to identify the closure of $A$ in the reals?
- 124
2
votes
0
answers
65
views
Request for resources or techniques for bounding the infinity norm of an infinite product convolved with a simple function
I'm attempting to bound an expression of the form.
$$
\lVert(\prod_{i=1}^{\infty} \phi_i) * s \rVert_{\infty}
$$
Where $\phi_i$ are bounded periodic step functions which can be replaced by smoothed ...
- 41
5
votes
0
answers
343
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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
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
3
votes
1
answer
302
views
number of integer points inside a triangle and its area
Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where
$\alpha, \beta, \gamma \in \mathbb{R}_{>0}$. I am interested in obtaining an estimate for the ...
- 3,625
5
votes
1
answer
351
views
Is every integer $\ge 312$ the sum of two integers with triangular divisors?
We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3$, $1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a ...
- 5,470
11
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1
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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
5
votes
1
answer
391
views
Proving a specific case of Robin's Inequality
Edit: It turns out that this is equivalent to the RH which gives the idea that this might a a little difficult to show. As such we could consider an even simpler case in which the number $n$ is ...
- 315
4
votes
1
answer
208
views
Stationary phase method for $\varphi''(x_0)= 0$
Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...
- 12.3k
4
votes
1
answer
247
views
Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?
Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
- 6,974
5
votes
0
answers
137
views
Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
- 5,470
6
votes
1
answer
274
views
Limits (growth rates) of power series coefficients
Take two positive integers $m$ and $n$ and consider the rational function
$$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G_{m,n}(x,t)=u_0(...
- 43.2k
28
votes
3
answers
3k
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Expressing the Riemann Zeta function in terms of GCD and LCM
Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
- 5,470
0
votes
1
answer
359
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Mistake in SageMathCell code, finding integral points on elliptic curves [closed]
I've the following number:
$$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$
Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has ...
- 111
0
votes
0
answers
81
views
equivalent of an alternating series
Let $d_n=\mathrm{lcm}(1,\cdots,n)$. By the prime number theorem $d_n=e^{n+o(n)}$.
I look for an equivalent of the function $\sum_{n\ge0}(-1)^n\frac{d_n}{n!}t^n$ when $t\to+\infty$. Unfortunately, the ...
- 3,998
6
votes
0
answers
267
views
Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
- 1,755
10
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0
answers
268
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On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$
I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well.
For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
- 377
1
vote
1
answer
526
views
A Related Problem to Erdős' similarity conjecture
Erdős' similarity conjecture states that for each infinite set $A\subset \mathbb R$ there is a set $P\subset [0,1]$ of positive measure such that for all $t\in \mathbb R$, $\delta\neq 0$ there is some ...
- 751
13
votes
3
answers
810
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
6
votes
1
answer
290
views
What is the growth rate of the products of binomial coefficients?
Question 1: Are the following empirically observed relationships true
$$
{n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a}
\sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)
$$
...
- 5,470
1
vote
0
answers
51
views
Mean value of a function with binomial coefficients as weights
Is the following true?
Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$
$$
\beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}...
- 5,470
5
votes
0
answers
89
views
Is the ratio of a number to the variance of its divisors injective?
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
- 5,470
2
votes
0
answers
249
views
Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral
It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$
and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$
where $\operatorname{li} (x)$ ...
0
votes
1
answer
177
views
Denominator approximation sequence of a real number
For any positive integer $[n]$, let $[n]=\{1,\ldots,n\}$. Let $r\in\mathbb{R}$. We define for every positive integer $n\in\mathbb{N}$ the minimal difference from a rational with denominator $\leq n$ ...
- 48.9k
2
votes
1
answer
228
views
Choosing finite subsets of natural numbers
Let $t>0$ and $\delta\in\big(0,\frac12\big)$ be fixed. For any $k\in\mathbb{N}$ let $I_k,J_k\in\mathbb{N}$ be finite subsets of natural numbers with cardinalities denoted as $|I_k|,|J_k|$, ...
- 375
1
vote
0
answers
291
views
An implication of the Zagier et al result on the hyperbolicity of Jensen polynomials for the Riemann zeta function?
In their paper recently published in the PNAS, Zagier et al demonstrated that
The Jensen polynomials $J_{\alpha}^{d,n}(X)$ of the Riemann zeta function of degree $d$ and shift $n$ are hyperbolic for ...
- 27
7
votes
1
answer
1k
views
Signed variant of the Flint Hills series
I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one:
$$
\sum_{n = 1}^\infty \...
- 17.2k
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
1
vote
0
answers
188
views
Questions on Riemann's explicit formula
If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
- 749
1
vote
1
answer
198
views
Comparison of two integrals
Let $S(x)$ be continuous, differentiable, and such that $S(x)=O(x/\log x)$. Let $J(x)=\int_x^{\infty} \frac{S(y)(1+\log y)}{y^2\log^2 y}dy$ and let $K(x)=\int_x^{\infty}\frac{S(y)}{y^2}dy$. Let $K(2)...
- 1,018