10
votes
Accepted
Estimate $ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n$ and $\sum_{\substack{n\leq x\\ n\in Q\\}} n$ where $Q$ is the square-free numbers
We can use the asymptotic formula
$$\displaystyle \sum_{\substack{n \leq x \\ n \in Q}} 1 = \frac{6x}{\pi^2} + O(x^{1/2}).$$
This asymptotic formula is very standard and is easy to prove. For a fixed ...
- 26.9k
5
votes
Accepted
What about a formula similar than Mill's formula, but producing positive integers without repeated prime factors?
Along the lines of the Wikipedia page, it is true that
$|Q(x)-\frac{x}{\zeta(2)}|\leq2+\sqrt{x}$
where $Q(x)$ is the number of square-free numbers between $1$ and $x$.
So,
$Q(n^3)\leq\frac{n^3}{\...
- 9,501
5
votes
Accepted
Conjectured error term when counting square-free integers
Your guess is correct! It is indeed conjectured that $a=1/4$. A good recent reference is [1]. In particular, it is known that
$$E(x)=\Omega(x^{1/4})$$
and computations have shown
$$|E(x)|<1.12543x^{...
- 465
5
votes
Accepted
Explicit bounds on number of squarefree numbers coprime to a certain number
I am assuming that explicit refers to the error term? In this case you can write $$Q_A(x)=\sum_{d\mid A}\mu(d)\sum_{k\leq \sqrt x \atop \gcd(A,k)=1}\mu(k)\left[\frac{x}{dk^2}\right],$$ where $[t]$ ...
- 3,062
4
votes
Distribution of $\{x/n^2\}$
I'm not sure but I'll point out two things which are hopefully somehow useful (and if they're not I'll just delete it later).
First, for the range close to $\sqrt x$: you have
\[ \int _{\epsilon \sqrt ...
- 1,381
4
votes
Radicands of square roots of the 2020s, written in simplest radical form
A simultaneous solution of $ax^2-c_1=b_1y^2$ and $ax^2-c_2=b_2z^2$ as demanded gives rise to an integral point on $b_1b_2T^2 = (ax^2-c_1)(ax^2-c_2)$, and since $a\ne 0$ and $c_1\ne c_2$, the degree-$4$...
- 2,959
3
votes
Estimate $ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n$ and $\sum_{\substack{n\leq x\\ n\in Q\\}} n$ where $Q$ is the square-free numbers
If $a_n = 1$ when $n$ is squarefree and $a_n = 0$ when $n$ is not squarefree, then you are asking for estimates on $\sum_{n \leq x} na_n$ and $\sum_{n \leq x} (\log n)a_n$. It is classical that $\...
- 50.6k
3
votes
Square-free numbers in an interval
Let me just show how to derive a simple bound that has been mentioned in the comments. We are trying to bound the estimate the number $Q(x,x+u)$ of squarefree integers in $(x,x+u]$.
We can now apply ...
- 20.2k
1
vote
Accepted
Justify that a certain set depending on a parameter is large
Let $p_1,\dots,p_s$ be the distinct primes in $t_1,\dots,t_k$.
We claim that, for any prime $p_i$, there is $a_i$ such that whenever $\ell \equiv a_i \pmod{p_i}$, we have $L - \ell^2 \not\equiv 0 \...
1
vote
Probabilistic interpretation of square free numbers and other properties
The standard way to formalize this thought is via the concept of "density".
This is because you of course cannot uniformly randomly select $n\in\mathbb{N}$, but you can instead examine (for a subset $...
- 2,183
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