Maximizing number of lattice points for Circles with real valued Radii

Question

Given an upper bound $L\in \mathbb{R}^*$, is there a tractable way to find a radius $r\leq L$ such that the number of lattice points on the circle centered at the origin with radius $r$ is maximized? As a followup, for what values of $r$ are we guaranteed to have a lattice point on the circle centered at the origin with radius $r$?

Answer 1

$$ r = \sqrt { 5 \cdot 13 \cdots p_k } $$ each prime $$ p_j \equiv 1 \pmod 4 $$

Interesting. I looked up Theorem 65 on page 80 of Dickson, Introduction to the Theory of Numbers, first published 1929. As $L$ gets large, we do the best by finding radii of a type of "highly composite" sort, that is $$ r = \sqrt { 5^{a_5} \cdot 13^{a_{13}} \cdot 17^{a_{17}} \cdots p_k^{a_{p_k}} }, $$ where $$ a_5 \geq a_{13} \geq a_{17} \geq \cdots \geq a_{p_k}, $$ and we demand that we have achieved the maximum possible of the product $$ (a_5 + 1) (a_{13} + 1) (a_{17} + 1)\cdots (a_{p_k} + 1)$$ such that $r < L.$ That is a fairly difficult calculation. Compare https://en.wikipedia.org/wiki/Highly_composite_number

Here are the first several "best" squared radii, you take the largest one permitted:

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   4  squared radius  1  =    1 
   8  squared radius  5  =   5
  12  squared radius  25  =   5^2
  16  squared radius  65  =   5 13
  24  squared radius  325  =   5^2 13
  32  squared radius  1105  =   5 13 17
  36  squared radius  4225  =   5^2 13^2
  48  squared radius  5525  =   5^2 13 17
  64  squared radius  27625  =   5^3 13 17
  72  squared radius  71825  =   5^2 13^2 17
  80  squared radius  138125  =   5^4 13 17
  96  squared radius  160225  =   5^2 13 17 29
 128  squared radius  801125  =   5^3 13 17 29
 144  squared radius  2082925  =   5^2 13^2 17 29
 160  squared radius  4005625  =   5^4 13 17 29
 192  squared radius  5928325  =   5^2 13 17 29 37
 256  squared radius  29641625  =   5^3 13 17 29 37
 288  squared radius  77068225  =   5^2 13^2 17 29 37
 320  squared radius  148208125  =   5^4 13 17 29 37
 384  squared radius  243061325  =   5^2 13 17 29 37 41
 512  squared radius  1215306625  =   5^3 13 17 29 37 41
 576  squared radius  3159797225  =   5^2 13^2 17 29 37 41
 640  squared radius  6076533125  =   5^4 13 17 29 37 41
 768  squared radius  12882250225  =   5^2 13 17 29 37 41 53
 864  squared radius  53716552825  =   5^2 13^2 17^2 29 37 41
1024  squared radius  64411251125  =   5^3 13 17 29 37 41 53
1152  squared radius  167469252925  =   5^2 13^2 17 29 37 41 53
1280  squared radius  322056255625  =   5^4 13 17 29 37 41 53
1536  squared radius  785817263725  =   5^2 13 17 29 37 41 53 61
1728  squared radius  2846977299725  =   5^2 13^2 17^2 29 37 41 53
2048  squared radius  3929086318625  =   5^3 13 17 29 37 41 53 61
2304  squared radius  10215624428425  =   5^2 13^2 17 29 37 41 53 61
2560  squared radius  19645431593125  =   5^4 13 17 29 37 41 53 61
3072  squared radius  51078122142125  =   5^3 13^2 17 29 37 41 53 61
3456  squared radius  173665615283225  =   5^2 13^2 17^2 29 37 41 53 61
3840  squared radius  255390610710625  =   5^4 13^2 17 29 37 41 53 61

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