Talk:Gauss circle problem

Cappell-Shaneson Paper?

There appears to be a paper by Cappell and Shaneson claiming to prove the $Cr^{1/2+\varepsilon }$ bound: it can be found on the ArXiv. Does anybody know if the paper has been reviewed for accuracy yet? Cpryby (talk) 22:03, 8 April 2009 (UTC)

Link to paper: http://arxiv.org/PS_cache/math/pdf/0702/0702613v3.pdf. Cpryby (talk) 22:04, 8 April 2009 (UTC)

It doesn't look too good: Chenxlee (talk) 17:24, 16 June 2011 (UTC)

Mohammad Ansari edit

Not that I am particularly knowledgeable on this topic, but $N(r)=1+4\sum _{k=0}^{[r]}{[{\sqrt {r^{2}-k^{2}}}]}.$ as added by Ansarimohammad does not look useful to me. Especially so as it was added without any reference. For this reason I have taken the liberty to remove it for the moment. _noisy jinx ^huh? 21:39, 30 October 2011 (UTC)

I think it was added to simplify programming the solution to this problem - although the equation by Ansarimohammad is invalid as the outcome will be short of the points lying on the X and Y axis, will be too big and won't be an integer as the ${\sqrt {r^{2}-k^{2}}}$ should've been floored. What he tried to say is $N(r)=1+4r+4\sum _{k=0}^{[r]}{\left\lfloor {\sqrt {r^{2}-k^{2}}}\right\rfloor }.$ I'm quite new to editing and consulting editions on Wiki but I don't think this formula requires any references - it just works, you can verify it programmatically or with pen and paper. I ran a java simulation for r=0..20000 and it does. It just means: 1. Count the origin point. 2. Count the points on the axis (it's a circle, so r to the right, left, up and down, hence 4r). 3. For every k between the origin and the edge of the circle check how many points there are vertically below the edge of the circle and above the X axis. It's a Pythagorean equation where r is hypotenus, k is one of the legs and square root of the difference of their squares is the other leg. Zegareke 01:25, 22 February 2017 (CEWT)

Can someone explain why this, or a variant of this, doesn't appear on the page? The formulae under the section "exact formulas" have either infinite or r^2 iterations. But it seems obvious that this is doable in O(r).

Last section, coprime problem

The last sentence states that an O(r^c) bound where c<1 cannot be proven without assuming the Riemann hypothesis. This means that proving such a bound, for example O(r^.9999), would prove the Riemann hypothesis. This is quite dubious to me (I am not an expert on this topic) but certainly has no citations. I'm rewording the sentence with this reasoning in mind; someone who knows better can revert. -Random user without a username

Yep, you're right. I meant to write "has not been proven without RH", not "can not be proven without RH." Thanks for correcting it! Chenxlee (talk) 16:47, 6 December 2011 (UTC)

Incorrect Gauss bound

The bound by Gauss is incorrect as stated: For $r\approx 0$ we have $N(r)=1$ and hence $E(r)\approx 1\gg 2{\sqrt {2}}\pi r$.--Hagman (talk) 17:05, 9 October 2013 (UTC)

You're correct. In addition, I read the claimed source and I didn't find this claim on page listed, or in the surrounding chapter about this problem. This source and other reliable sources I've seen claim that Gauss proved $E(r)=O(r)$, but I haven't seen any that make this exact claim.

I've changed the citation to "citation needed" and the inequality to the result of expanding $\pi (r-{\sqrt {2}})^{2}\leq N(r)\leq \pi (r+{\sqrt {2}})^{2}$, but an actual source would be nice. Vpmovmskb (talk) 05:15, 14 April 2026 (UTC)

number fields

The problem generalizes to arbitrary number fields, and there is apparently a literature about the problem in that setting, in case anyone wants to use it in the article:

• MO thread

173.228.123.101 (talk) 07:49, 19 December 2015 (UTC)

Bad citation to the conjecture

The article says that it is conjectured that $|E(r)| \leq O(r^{1/2 + \varepsilon}$ and gives a citation to Chapter F1 of Guy's book. But, I did not find this conjecture there. NoahSD (talk) 22:23, 26 November 2024 (UTC)