Sums of two squares in arithmetic progressions












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$begingroup$


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






nt.number-theory reference-request arithmetic-progression sums-of-squares







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  • $begingroup$
    If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
    $endgroup$
    – Dongryul Kim
    3 hours ago
















4












$begingroup$


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






nt.number-theory reference-request arithmetic-progression sums-of-squares







share|cite|improve this question











$endgroup$












  • $begingroup$
    If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
    $endgroup$
    – Dongryul Kim
    3 hours ago














4












4








4


1



$begingroup$


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






nt.number-theory reference-request arithmetic-progression sums-of-squares







share|cite|improve this question











$endgroup$




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






nt.number-theory reference-request arithmetic-progression sums-of-squares




nt.number-theory reference-request arithmetic-progression sums-of-squares






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share|cite|improve this question













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edited 2 hours ago









Martin Sleziak

3,09032231




3,09032231










asked 3 hours ago









cawscaws

462




462












  • $begingroup$
    If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
    $endgroup$
    – Dongryul Kim
    3 hours ago


















  • $begingroup$
    If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
    $endgroup$
    – Dongryul Kim
    3 hours ago
















$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
3 hours ago




$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
3 hours ago










1 Answer
1






active

oldest

votes


















7












$begingroup$

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
$$sum _{nleq xatop {nequiv a(q)}}r(n) =pi x cdot frac{eta_{a}(q)}{q^2}+ R_{q,a}(x)$$
where $eta_{a}(q) := { (x_1,x_2) in (mathbb{Z}/qmathbb{Z)}^2 : x_1^2 +x_2^2 equiv a bmod q}$, then
$$R_{q,a}(x) = Oleft( x^{frac{2}{3} + xi} q^{-frac{1}{2}(1+3xi)}gcd(a,q)^{1/2}tau(q) right)$$
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
$$R_{q,a}(x) = Oleft( (q^{frac{1}{2}}+x^{frac{1}{3}}) gcd(a,q)^{1/2}tau^4(q)log^4 x right).$$
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)].



All three results are explained in Tolev's paper.




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
$$f(q,a):=lim_{x to infty} frac{sum _{nleq xatop {nequiv a(q)}}r(n) }{pi x}$$
exists and can be written as
$$f(q,a)=q^{-3} sum_{k=1}^{q} expleft( 2pi i frac{-ak}{q} right) S(q,k)^2,$$
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
$$R_{q,a}(x) = Oleft( (sqrt{x}+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.






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    $begingroup$

    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
    $$sum _{nleq xatop {nequiv a(q)}}r(n) =pi x cdot frac{eta_{a}(q)}{q^2}+ R_{q,a}(x)$$
    where $eta_{a}(q) := { (x_1,x_2) in (mathbb{Z}/qmathbb{Z)}^2 : x_1^2 +x_2^2 equiv a bmod q}$, then
    $$R_{q,a}(x) = Oleft( x^{frac{2}{3} + xi} q^{-frac{1}{2}(1+3xi)}gcd(a,q)^{1/2}tau(q) right)$$
    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
    $$R_{q,a}(x) = Oleft( (q^{frac{1}{2}}+x^{frac{1}{3}}) gcd(a,q)^{1/2}tau^4(q)log^4 x right).$$
    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)].



    All three results are explained in Tolev's paper.




    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
    $$f(q,a):=lim_{x to infty} frac{sum _{nleq xatop {nequiv a(q)}}r(n) }{pi x}$$
    exists and can be written as
    $$f(q,a)=q^{-3} sum_{k=1}^{q} expleft( 2pi i frac{-ak}{q} right) S(q,k)^2,$$
    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
    $$R_{q,a}(x) = Oleft( (sqrt{x}+q) log qright).$$
    For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.






    share|cite|improve this answer











    $endgroup$


















      7












      $begingroup$

      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
      $$sum _{nleq xatop {nequiv a(q)}}r(n) =pi x cdot frac{eta_{a}(q)}{q^2}+ R_{q,a}(x)$$
      where $eta_{a}(q) := { (x_1,x_2) in (mathbb{Z}/qmathbb{Z)}^2 : x_1^2 +x_2^2 equiv a bmod q}$, then
      $$R_{q,a}(x) = Oleft( x^{frac{2}{3} + xi} q^{-frac{1}{2}(1+3xi)}gcd(a,q)^{1/2}tau(q) right)$$
      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
      $$R_{q,a}(x) = Oleft( (q^{frac{1}{2}}+x^{frac{1}{3}}) gcd(a,q)^{1/2}tau^4(q)log^4 x right).$$
      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)].



      All three results are explained in Tolev's paper.




      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
      $$f(q,a):=lim_{x to infty} frac{sum _{nleq xatop {nequiv a(q)}}r(n) }{pi x}$$
      exists and can be written as
      $$f(q,a)=q^{-3} sum_{k=1}^{q} expleft( 2pi i frac{-ak}{q} right) S(q,k)^2,$$
      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
      $$R_{q,a}(x) = Oleft( (sqrt{x}+q) log qright).$$
      For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.






      share|cite|improve this answer











      $endgroup$
















        7












        7








        7





        $begingroup$

        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
        $$sum _{nleq xatop {nequiv a(q)}}r(n) =pi x cdot frac{eta_{a}(q)}{q^2}+ R_{q,a}(x)$$
        where $eta_{a}(q) := { (x_1,x_2) in (mathbb{Z}/qmathbb{Z)}^2 : x_1^2 +x_2^2 equiv a bmod q}$, then
        $$R_{q,a}(x) = Oleft( x^{frac{2}{3} + xi} q^{-frac{1}{2}(1+3xi)}gcd(a,q)^{1/2}tau(q) right)$$
        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
        $$R_{q,a}(x) = Oleft( (q^{frac{1}{2}}+x^{frac{1}{3}}) gcd(a,q)^{1/2}tau^4(q)log^4 x right).$$
        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)].



        All three results are explained in Tolev's paper.




        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
        $$f(q,a):=lim_{x to infty} frac{sum _{nleq xatop {nequiv a(q)}}r(n) }{pi x}$$
        exists and can be written as
        $$f(q,a)=q^{-3} sum_{k=1}^{q} expleft( 2pi i frac{-ak}{q} right) S(q,k)^2,$$
        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
        $$R_{q,a}(x) = Oleft( (sqrt{x}+q) log qright).$$
        For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.






        share|cite|improve this answer











        $endgroup$



        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
        $$sum _{nleq xatop {nequiv a(q)}}r(n) =pi x cdot frac{eta_{a}(q)}{q^2}+ R_{q,a}(x)$$
        where $eta_{a}(q) := { (x_1,x_2) in (mathbb{Z}/qmathbb{Z)}^2 : x_1^2 +x_2^2 equiv a bmod q}$, then
        $$R_{q,a}(x) = Oleft( x^{frac{2}{3} + xi} q^{-frac{1}{2}(1+3xi)}gcd(a,q)^{1/2}tau(q) right)$$
        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
        $$R_{q,a}(x) = Oleft( (q^{frac{1}{2}}+x^{frac{1}{3}}) gcd(a,q)^{1/2}tau^4(q)log^4 x right).$$
        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)].



        All three results are explained in Tolev's paper.




        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
        $$f(q,a):=lim_{x to infty} frac{sum _{nleq xatop {nequiv a(q)}}r(n) }{pi x}$$
        exists and can be written as
        $$f(q,a)=q^{-3} sum_{k=1}^{q} expleft( 2pi i frac{-ak}{q} right) S(q,k)^2,$$
        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
        $$R_{q,a}(x) = Oleft( (sqrt{x}+q) log qright).$$
        For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 hours ago

























        answered 2 hours ago









        Ofir GorodetskyOfir Gorodetsky

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