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The numbers of lattice points falling on the circumference of circles centered at the origin of radii $0, 1, 2, \ldots$ are known to be $$1, 4, 4, 4, 4, 12, 4, 4, 4, 4, 12, 4, 4,\ldots$$ see on the Online Encyclopeadia of Integer Sequences (OEIS A046109).

The relevant function is $r_2(n^2)$ or the sum of squares function. More generally $r_k(n)$ is the function counting the number of representations of $n$ by $k$ squares, allowing zeros and distinguishing signs and order. See equations (17) and (18) in https://mathworld.wolfram.com/SumofSquaresFunction.html.

See also, https://mathworld.wolfram.com/CircleLatticePoints.html for more details.

Is it known whether one can get more lattice points on a circle centred at origin if radii that are not nonnegative integers are allowed? It seems like the answer should be "no".

What about the maximum

$$M(n):=\max\{ r_2(x) : x \in (0,n] \}?$$

Is it possible to bound this maximum from above? It should be less than the maxima achieved on the integers as in the above OEIS sequence.

Jinyuan
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kodlu
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    The sequence you list at the start is $r_2(n^2)$, not $r_2(n)$. So your question is just about $r_2(n)$ when $n$ is not a perfect square. It is easy to see from formula (18) that $r_2(n)$ does not always reach its new "record highs" at perfect square values (for instance, $r_2(65)=16$ is a record high since $65$ is the first number with two distinct prime factors that are 1 mod 4 and is smaller than $5^3$.). – Eric Wofsey Jul 22 '22 at 20:22
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    See https://math.stackexchange.com/questions/2349068/which-numbers-less-than-5-billion-have-the-most-representations-as-the-sums-of-t In brief, there is a sequence of "record high" numbers very similar to Ramanujan's Superior Highly Composite numbers, in that there is a simple recipe to construct them. Other numbers may set new record highs, these will have factorizations fairly similar to the recipe.. – Will Jagy Jul 23 '22 at 01:38

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