Sums of two squares in arithmetic progressions
$begingroup$
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 analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5148227
asked Apr 2 at 12:30
cawscaws
734
$endgroup$
add a comment |
$begingroup$
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 analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5148227
asked Apr 2 at 12:30
cawscaws
734
$endgroup$
$begingroup$
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
Apr 2 at 12:50
$begingroup$
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.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
add a comment |
$begingroup$
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 analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5148227
asked Apr 2 at 12:30
cawscaws
734
$endgroup$
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 analytic-number-theory arithmetic-progression sums-of-squares
nt.number-theory reference-request analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5148227
asked Apr 2 at 12:30
cawscaws
734
edited Apr 2 at 19:33
GH from MO
59.3k5148227
asked Apr 2 at 12:30
cawscaws
734
edited Apr 2 at 19:33
GH from MO
59.3k5148227
edited Apr 2 at 19:33
GH from MO
59.3k5148227
edited Apr 2 at 19:33
GH from MO
59.3k5148227
59.3k5148227
asked Apr 2 at 12:30
cawscaws
734
asked Apr 2 at 12:30
cawscaws
734
asked Apr 2 at 12:30
cawscaws
734
734
$begingroup$
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
Apr 2 at 12:50
$begingroup$
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.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
add a comment |
$begingroup$
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
Apr 2 at 12:50
$begingroup$
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.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
$begingroup$
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
Apr 2 at 12:50
$begingroup$
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
Apr 2 at 12:50
$begingroup$
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.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
$begingroup$
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.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
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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.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
$begingroup$
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.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
$begingroup$
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.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
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.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
edited Apr 2 at 14:08
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
5,90312639
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
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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
Apr 2 at 12:50
$begingroup$
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.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10