Sums of two squares in arithmetic progressionsSums of two squares in (certain) integral domainsIs there another simple formula for the sum-of-squares function?Exact formula for the number of integers in an interval which are the sum of two squares.Arithmetic Progressions of SquaresSums of two squares: What is known about the distribution of r(n)?Primes in arithmetic progressions in number fieldsJacobi's theorem on sums of two squares (reference request)Average class number formula for imaginary quadratic fields with prime discriminantSums of four coprime squaresSums of two integer squares in arithmetic progressions
Sums of two squares in arithmetic progressions
Sums of two squares in (certain) integral domainsIs there another simple formula for the sum-of-squares function?Exact formula for the number of integers in an interval which are the sum of two squares.Arithmetic Progressions of SquaresSums of two squares: What is known about the distribution of r(n)?Primes in arithmetic progressions in number fieldsJacobi's theorem on sums of two squares (reference request)Average class number formula for imaginary quadratic fields with prime discriminantSums of four coprime squaresSums of two integer 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 2 days ago
GH from MO
59.2k5148227
asked 2 days ago
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 2 days ago
GH from MO
59.2k5148227
asked 2 days ago
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
2 days ago
$begingroup$
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
2 days ago
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 2 days ago
GH from MO
59.2k5148227
asked 2 days ago
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 2 days ago
GH from MO
59.2k5148227
asked 2 days ago
cawscaws
734
edited 2 days ago
GH from MO
59.2k5148227
asked 2 days ago
cawscaws
734
edited 2 days ago
GH from MO
59.2k5148227
edited 2 days ago
GH from MO
59.2k5148227
edited 2 days ago
GH from MO
59.2k5148227
59.2k5148227
asked 2 days ago
cawscaws
734
asked 2 days ago
cawscaws
734
asked 2 days ago
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
2 days ago
$begingroup$
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
2 days ago
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
2 days ago
$begingroup$
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
2 days 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
2 days 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
2 days ago
$begingroup$
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
2 days ago
$begingroup$
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
2 days ago
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
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. 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
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
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
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
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
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
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
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. 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
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
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
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
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
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
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
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. 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
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
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
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
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
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
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
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. 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
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
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
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
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
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered 2 days ago
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
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. 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
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
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
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
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
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
edited 2 days ago
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
answered 2 days ago
Ofir GorodetskyOfir Gorodetsky
5,90312639
5,90312639
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
add a comment |
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
2 days ago
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
2 days ago
$begingroup$
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
2 days ago