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 arithmetic-progression sums-of-squares
edited 2 hours ago
Martin Sleziak
3,09032231
asked 3 hours ago
cawscaws
462
$endgroup$
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 arithmetic-progression sums-of-squares
edited 2 hours ago
Martin Sleziak
3,09032231
asked 3 hours ago
cawscaws
462
$endgroup$
$begingroup$
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
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 arithmetic-progression sums-of-squares
edited 2 hours ago
Martin Sleziak
3,09032231
asked 3 hours ago
cawscaws
462
$endgroup$
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
edited 2 hours ago
Martin Sleziak
3,09032231
asked 3 hours ago
cawscaws
462
edited 2 hours ago
Martin Sleziak
3,09032231
asked 3 hours ago
cawscaws
462
edited 2 hours ago
Martin Sleziak
3,09032231
edited 2 hours ago
Martin Sleziak
3,09032231
edited 2 hours ago
Martin Sleziak
3,09032231
3,09032231
asked 3 hours ago
cawscaws
462
asked 3 hours ago
cawscaws
462
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
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
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
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 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f326964%2fsums-of-two-squares-in-arithmetic-progressions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
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 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
$endgroup$
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 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
$endgroup$
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 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
$endgroup$
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 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
edited 2 hours ago
answered 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
answered 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
answered 2 hours ago
Ofir GorodetskyOfir Gorodetsky
5,79812538
5,79812538
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f326964%2fsums-of-two-squares-in-arithmetic-progressions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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