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Results tagged with nt.number-theory
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user 342938
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
18
votes
1
answer
589
views
For which $n$ is $\sum_{k=1}^n 1 / \varphi(k)$ an integer?
For which positive integers $n$ is the sum $\sum_{k=1}^n 1 / \varphi(k)$ an integer? Here $\varphi$ is the Euler totient function.
The question is a "totient-analog" of the well-known result that $\su …
- 453
14
votes
1
answer
691
views
Numbers without prime factors in a set of positive relative density
Let $\mathcal{P}$ be a set of prime numbers of relative density $\kappa \in (0, 1)$, which means that
$$\#\left(\mathcal{P} \cap [1, x]\right) = \kappa \,\pi(x) + E(x) \quad (x \to \infty)$$
for a "su …
- 453
3
votes
0
answers
115
views
p-adic density of the image of a polynomial
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. Recently, a user of MO proved that the limit
$$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\ …
- 453