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Results tagged with nt.number-theory
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user 169474
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
1
vote
Natural extension of the Gauss map
Partial answer.
I imagine that you mean a map $\overline{G}$ which sends $([0;a_1,a_2,\ldots],[0;a_0,a_{-1},\ldots])$ on $([0;a_2,a_3,\ldots],[0;a_1,a_0,\ldots])$. If yes, we have a simple formula
$$\ …
- 3,366
7
votes
Square root of prime numbers
My impression is that you have developed $(x_0-\sqrt{S})^{2^n}$ with Newton binomial formula, separated the terms with even index from the terms with odd index to get an expression of the form $A_n-\s …
- 3,366
1
vote
Growth of the "cube of square root" function
For $-1<h<1$,
$$(1+h)^{3/2}+(1-h)^{3/2} = 2 \sum_{n=0}^{+\infty}{3/2 \choose 2n}h^{2n},$$
where
$${3/2 \choose 2n} = \prod_{k=1}^ {2n} \frac{5/2-k}{k}.$$
For $n \ge 1$, since $(-1)^{2n-2}=1$, we get
$ …
- 3,366
7
votes
Accepted
Central limit theorem for irrational rotations
I am surprised by the question. The sums are bounded if $\alpha \ne 1$, since for every $n \ge 1$,
$$\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big|
= \Big|\Re \Big(\sum_{k=1}^n R_\alpha^k z \Big) \Big| = \B …
- 3,366
1
vote
Accepted
Convergence and roots of alternating periodic infinite series
I prove the convergence of the series.
For $n \ge 1$, let
$$S_n = \sum_{k=1}^n k^{-i\beta-\alpha} \text{ and } S'_n = \sum_{k=1}^n (-1)^ {k-1} k^{-i\beta-\alpha}$$
Then
$$S_{2n}-S'_{2n} = 2 \sum_{k=1} …
- 3,366