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The original question was: In my recent article (see section 2.3 here) I claim that the number of solutions to $x^2 + y^2 \leq k$ with unknown $(x, y)$ being integers (positive or negative) is asymptotically $\pi k$ as $k$ tends to infinity. Is this correct, is my reasoning correct in my article?

Updated question: Since a simple answer was provided by Daniel Fisher (this is the Gauss circle problem), I have now two deeper questions. One for which I believe I have an answer based on heuristic (see below), and one for which a reference may exist. These questions are related to the number of solutions to $x^2 + y^2 =n$ as $n$ gets very large. That number is denoted as $r(n)$, see here.

  • Question 1: What is the asymptotic rate of growth for successive records of $r(n)$, as a function of $n$?
  • Question 2: What is the rate of growth (asymptotically, on average) for $r(n)$, as a function of $n$ and only for values of $n$ such that $r(n) > 0$?

I may have an answer for question 2, and I am wondering if it can be made rigorous. It is backed by empirical evidence. Here is my approach. For $k$ between $n$ and $2n$, we have about $$\frac{2cn}{\sqrt{\log 2n}} - \frac{cn}{\sqrt{\log n}} \sim \frac{cn}{\sqrt{\log n}}$$ integers such that $r(k)>0$, based on this well known result, where $c=0.7642...$ is the Landau-Ramanujan constant. Also based on my initial question (the Gauss circle problem) the total number of solutions across all $k$'s between $n$ and $2n$ is asymptotically $2\pi n - \pi n = \pi n$. These $\pi n$ solutions are distributed across $cn/\sqrt{\log n}$ integers satisfying $r(k) > 0$. This means that for large $n$'s satisfying $r(n)>0$, the average value of $r(n)$ grows as $\pi \sqrt{\log n}/c$.

A consequence of this is that the records for $r(n)$ grow faster than $\pi \sqrt{\log n}/c$. How much faster? By a constant? That, I don't know.

Note: Sums of squares are far more abundant than primes, yet their natural density is also zero. This means that as $n$ becomes larger and larger, the chance for $n$ to be a sum of two squares, tends to zero. But for the few remaining $n$'s that can be expressed as a sum of two squares, the number of ways they can be expressed that way, increases more and more on average. That is, $r(n)$ tends to increase with $n$, for those few large $n$ satisfying $r(n) > 0$. As a result, for any $n$ large or small, the number of solutions to $x^2 + y^2 = n$ is equal to $\pi$ on average. What changes (it increases) when $n$ is large is the probability that $r(n) = 0$, but the average value of $r(n)$ remains unchanged.

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    $\begingroup$ en.wikipedia.org/wiki/Gauss_circle_problem $\endgroup$ Jun 5, 2020 at 20:18
  • $\begingroup$ @DanielFischer: Thank you for you answer! I updated my question, I believe it is now much more interesting. $\endgroup$ Jun 6, 2020 at 13:49
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    $\begingroup$ Mathworld gives a formula for $r(n)$ based on the prime factorization of $n$. The records of $r(n)$ will come when $n$ has the correct type of factorization. It is related to, but different from, highly composite numbers because primes of the form $4k+3$ are penalized by having to come with even exponents. $\endgroup$ Jun 6, 2020 at 13:55

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