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What is the status of the Gauss Circle Problem?

The Gauss circle problem is the following: Let $N(r)$ denote the number of solutions in integer pairs $(i,j)$ to the inequality $i^2 + j^2 \leq r^2$ (namely, the number of lattice points on or inside the disc centered at 0 with radius $r$). Then it is easy to see we should have $N(r) = \pi r^2 + E(r)$ for some error term $E(r)$, where the key is to estimate $E(r)$. Gauss proved that $E(r) \leq 2\sqrt{2}\pi r$, and Landau showed that $E(r) \ne o(r^{1/2}\log^{1/4}(r))$. The conjecture is that $E(r) = O(r^{1/2 + \epsilon})$ for any $\epsilon > 0$. If the conjecture is true, then the squares will provide an explicit example of a subset of positive integers $A$ such that that the representation function $r_A(n)$ defined to be the number of ways of writing $n$ as a sum of two elements of $A$ satisfies $\displaystyle \sum_{j \leq n} r_A(j) = cn + O(n^{1/4}\log(n))$. Such sets $A$ exist by a result of I. Ruzsa in 1999, but no known examples exist.

So my question is, what is the best known result on the Gauss circle problem? Or in lieu of that, a good explanation on why this problem is so difficult?

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It's also interesting (although you probably already know this) to look at the Dirichlet Divisor Problem. I believe, although I am not an expert in this area, that the two problems have many similar methods and results; this is not surprising, since it involves the number of lattice points under a hyperbola, instead of a circle as here. – Zen Harper Jan 26 '11 at 19:57
Although this overlaps with the question that Dror links to, my first impression is that the question being asked here has a different aim or focus, so I think it deserves to stay open. – Yemon Choi Jan 26 '11 at 22:28
I was a little too quick on the trigger on closing this one, and afterwards thought better of it (I think Yemon has a point). Unfortunately, this had the effect of wiping out one closing vote, but I guess we can consider that one as canceled by Yemon's vote to stay open. – Ben Webster Jan 27 '11 at 7:45
Just ressurecting the question, is there any estimate on the constants ? I mean, if I wanted to apply these results, I would like to know C such that $|E| < C r^\theta$.. – Campello Aug 12 '14 at 3:19
up vote 5 down vote accepted

The best reference on the subject is M. Huxley's monograph

MR1420620 (97g:11088) Huxley, M. N.(4-WALC) Area, lattice points, and exponential sums. London Mathematical Society Monographs. New Series, 13. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xii+494 pp. ISBN: 0-19-853466-3 11L07 (11J54 11P21)

The proof of the circle conjecture has been claimed by Cappell/Shaneson, but I haven't read their paper, don't know anyone who has, so cannot comment on its correctness (maybe someone else here can).

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See A. Ivic, E. Kratzel, M. Kuhleitner, W. G. Nowak, Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic, Elementare und analytische Zahlentheorie, 89–128, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006, MR2310176 (2008b:11105).

The review, written by Wolfgang Schwarz, begins, "This paper is an up-to-date (2004) and most interesting survey paper (with an emphasis on results established during the last few years) on estimates of the number of lattice points in large regions, with nearly 200 references.

"Section 1 deals with the Gauss' circle problem (Hardy's identity, O-estimates, lower bounds, mean-square estimates)."

Some other recent papers that might be of interest:

Martine Babillot, Points entiers et groupes discrets: de l'analyse aux systemes dynamiques, Panor. Syntheses, 13, Rigidite, groupe fondamental et dynamique, 1–119, Soc. Math. France, Paris, 2002. MR1993148 (2004i:37057)

Fernando Chamizo, Antonio Cordoba, Lattice points, Margarita mathematica, 59–76, Univ. La Rioja, Logroño, 2001. MR1882616 (2003j:11118)

M. N. Huxley, Integer points in plane regions and exponential sums, Number theory, 157–166, Trends Math., Birkhäuser, Basel, 2000. MR1764801 (2002i:11101)

Wenguang Zhai, On higher-power moments of Δ(x). III, Acta Arith. 118 (2005), no. 3, 263–281. MR2168766 (2006f:11121)

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