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The distribution of sums of two squares has been studied by Landau. What is known about the distribution of the function $r(n)$, the number of representations of $n$ as the sum of two squares? Some specific questions of interest to us are these. Suppose $a \leq n\leq b$. What is the average value of $r(n)$? What is the standard deviation?

Edit: Thanks for the answer and comments. I am particularly interested in the range $a\leq n\leq b$ where $b-a = \Theta(\sqrt{a})$. Let $a,b$ be like that, and let $C(n)$ be the circle of radius $n$ around a fixed center.

Conjecture. The number of integers $n\in[a,b]$ with $r(n)>0$ is $\Omega(\sqrt{a})$. Furthermore, if $S$ is the segment of $C(b)$ cut off by a chord of length $\Omega(\sqrt{a})$ that touches $C(a)$ and if $r_S(n)$ is the number of lattice points in $C(n)\cap S$ then the number of integers $n\in[a,b]$ with $r_S(n)>0$ is $\Omega(\sqrt{a})$.

The distribution of sums of two squares has been studied by Landau. What is known about the distribution of the function $r(n)$, the number of representations of $n$ as the sum of two squares? Some specific questions of interest to us are these. Suppose $a \leq n\leq b$. What is the average value of $r(n)$? What is the standard deviation?

Edit: Thanks for the answer and comments. I am particularly interested in the range $a\leq n\leq b$ where $b-a = \Theta(\sqrt{a})$. Let $a,b$ be like that, and let $C(n)$ be the circle of radius $n$ around a fixed center.

Conjecture. The number of integers $n\in[a,b]$ with $r(n)>0$ is $\Omega(\sqrt{a})$. Furthermore, if $S$ is the segment of $C(b)$ cut off by a chord of length $\Omega(\sqrt{a})$ that touches $C(a)$ and if $r_S(n)$ is the number of lattice points in $C(n)\cap S$ then the number of integers $n\in[a,b]$ with $r_S(n)>0$ is $\Omega(\sqrt{a})$.

Added: Finally, with the help of Greg Martin, I got hold of relevant literature and educated myself. Thank you again for the useful answers and comments!

The distribution of sums of two squares has been studied by Landau. What is known about the distribution of the function $r(n)$, the number of representations of $n$ as the sum of two squares? Some specific questions of interest to us are these. Suppose $a \leq n\leq b$. What is the average value of $r(n)$? What is the standard deviation?

Edit: Thanks for the answer and comments. I am particularly interested in the range $a\leq n\leq b$ where $b$ is about $a+\sqrt{a}$.

The distribution of sums of two squares has been studied by Landau. What is known about the distribution of the function $r(n)$, the number of representations of $n$ as the sum of two squares? Some specific questions of interest to us are these. Suppose $a \leq n\leq b$. What is the average value of $r(n)$? What is the standard deviation?

Edit: Thanks for the answer and comments. I am particularly interested in the range $a\leq n\leq b$ where $b-a = \Theta(\sqrt{a})$. Let $a,b$ be like that, and let $C(n)$ be the circle of radius $n$ around a fixed center.

Conjecture. The number of integers $n\in[a,b]$ with $r(n)>0$ is $\Omega(\sqrt{a})$. Furthermore, if $S$ is the segment of $C(b)$ cut off by a chord of length $\Omega(\sqrt{a})$ that touches $C(a)$ and if $r_S(n)$ is the number of lattice points in $C(n)\cap S$ then the number of integers $n\in[a,b]$ with $r_S(n)>0$ is $\Omega(\sqrt{a})$.