# Sums of two squares in arithmetic progressions

Let $$r(n)$$ denote the number of representations of $$n$$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic formula?

• If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice? – Dongryul Kim Apr 2 '19 at 12:50
• It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques. – Daniel Loughran Apr 2 '19 at 19:10

The first result in this direction seems to be due to R. A. Smith, The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n) =\pi x \cdot \frac{\eta_{a}(q)}{q^2}+ R_{q,a}(x)$$ where $$\eta_{a}(q) := \{ (x_1,x_2) \in (\mathbb{Z}/q\mathbb{Z)}^2 : x_1^2 +x_2^2 \equiv a \bmod q\}$$, then $$R_{q,a}(x) = O\left( x^{\frac{2}{3} + \xi} q^{-\frac{1}{2}(1+3\xi)}\gcd(a,q)^{1/2}\tau(q) \right)$$ for any $$\xi \in (0,1/3)$$. This is non-trivial for $$q \le x^{\frac{2}{3}-\varepsilon}$$. In particular, as $$x$$ tends to infinity, your expression is asymptotic to $$\pi x$$ times the probability that $$x_1^2+x_2^2 \equiv a \bmod q$$ (for uniformly drawn $$x_1,x_2$$ mod $$q$$). If you consider $$a$$ and $$q$$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that $$R_{q,a}(x) = O\left( (q^{\frac{1}{2}}+x^{\frac{1}{3}}) \gcd(a,q)^{1/2}\tau^4(q)\log^4 x \right).$$ Interestingly, for $$a=1$$, there is a result which is superior in certain ranges of $$x$$ and $$q$$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that $$f(q,a):=\lim_{x \to \infty} \frac{\sum _{n\leq x\atop {n\equiv a(q)}}r(n) }{\pi x}$$ exists and can be written as $$f(q,a)=q^{-3} \sum_{k=1}^{q} \exp\left( 2\pi i \frac{-ak}{q} \right) S(q,k)^2,$$ where $$S(q,a)$$ is a quadratic Gauss sum mod $$q$$. This gives a different expression for $$\eta_a(q)/q^2$$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate $$R_{q,a}(x) = O\left( (\sqrt{x}+q) \log q\right).$$ For small $$q$$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.