All Questions
Tagged with nt.number-theory real-analysis
59 questions with no upvoted or accepted answers
0
votes
0
answers
71
views
Nearest integer to fractional power series
Let $k$ be a positive integer. Let
$$\displaystyle f_0(x) = a_n x^{\frac{n}{k}} + \cdots + a_1 x^{\frac{1}{k}} + a_0 + \sum_{h \geq 1} a_{-h} x^{-\frac{h}{k}}$$
be a Laurent series in the variable $x^{...
- 26.9k
0
votes
0
answers
63
views
Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 47
0
votes
0
answers
122
views
Convergence of a series related to counting distinct prime factors
I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
0
votes
0
answers
107
views
$\log$-classes of irrationals
Let $\mathbb{N}$ denote the set of non-negative integers. For $A\subseteq \mathbb{N}$ we let the (upper) density of $A$ be defined by $d^+(A) = \lim\sup_{n\to\infty} \frac {|A\cap \{0,\ldots, n\}|}{n+...
- 48.9k
0
votes
0
answers
82
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
0
votes
0
answers
221
views
Positivity of an integral
Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...
- 1,018
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:
$$D(X)=\lim_{r\rightarrow \infty} \inf_{y\in\...
- 297
-1
votes
1
answer
173
views
For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$
Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
- 11
-2
votes
1
answer
209
views
Strong estimates for the zeta function on natural numbers
Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$
be the Riemann zeta function (here we just consider real $s$).
We do have a description given by
$$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
- 749