Does bounded version of Lagrange's 4-square theorem follow from this result?
$begingroup$
Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $sum_{i=1}^nS_4(i) sim frac{pi^2}{2}n^2$.
Is this result enough to deduce that beyond some $N$, every integer can be represented as the sum of 4 squares (given that we are "weakly" suggesting that $sum_{i=1}^nS_4(i) >> n$)?
number-theory proof-verification asymptotics sums-of-squares
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
$endgroup$
add a comment |
$begingroup$
Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $sum_{i=1}^nS_4(i) sim frac{pi^2}{2}n^2$.
Is this result enough to deduce that beyond some $N$, every integer can be represented as the sum of 4 squares (given that we are "weakly" suggesting that $sum_{i=1}^nS_4(i) >> n$)?
number-theory proof-verification asymptotics sums-of-squares
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
$endgroup$
1
$begingroup$
It's not true that $S_4(n)simfrac{pi^2}{2}n^2$. What is true is that $sum_{k=1}^nS_4(k)simfrac{pi^2}{2}n^2$.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 16:16
$begingroup$
Ah yes, I meant that! Let me just update this.
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:12
$begingroup$
@LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual...
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:13
$begingroup$
You need more than that asymptotic.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 19:13
add a comment |
$begingroup$
Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $sum_{i=1}^nS_4(i) sim frac{pi^2}{2}n^2$.
Is this result enough to deduce that beyond some $N$, every integer can be represented as the sum of 4 squares (given that we are "weakly" suggesting that $sum_{i=1}^nS_4(i) >> n$)?
number-theory proof-verification asymptotics sums-of-squares
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
$endgroup$
Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $sum_{i=1}^nS_4(i) sim frac{pi^2}{2}n^2$.
Is this result enough to deduce that beyond some $N$, every integer can be represented as the sum of 4 squares (given that we are "weakly" suggesting that $sum_{i=1}^nS_4(i) >> n$)?
number-theory proof-verification asymptotics sums-of-squares
number-theory proof-verification asymptotics sums-of-squares
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
edited Dec 15 '18 at 21:39
Isky Mathews
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
asked Dec 15 '18 at 16:13
Isky MathewsIsky Mathews
868314
868314
1
$begingroup$
It's not true that $S_4(n)simfrac{pi^2}{2}n^2$. What is true is that $sum_{k=1}^nS_4(k)simfrac{pi^2}{2}n^2$.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 16:16
$begingroup$
Ah yes, I meant that! Let me just update this.
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:12
$begingroup$
@LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual...
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:13
$begingroup$
You need more than that asymptotic.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 19:13
add a comment |
1
$begingroup$
It's not true that $S_4(n)simfrac{pi^2}{2}n^2$. What is true is that $sum_{k=1}^nS_4(k)simfrac{pi^2}{2}n^2$.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 16:16
$begingroup$
Ah yes, I meant that! Let me just update this.
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:12
$begingroup$
@LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual...
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:13
$begingroup$
You need more than that asymptotic.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 19:13
1
1
$begingroup$
It's not true that $S_4(n)simfrac{pi^2}{2}n^2$. What is true is that $sum_{k=1}^nS_4(k)simfrac{pi^2}{2}n^2$.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 16:16
$begingroup$
It's not true that $S_4(n)simfrac{pi^2}{2}n^2$. What is true is that $sum_{k=1}^nS_4(k)simfrac{pi^2}{2}n^2$.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 16:16
$begingroup$
Ah yes, I meant that! Let me just update this.
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:12
$begingroup$
Ah yes, I meant that! Let me just update this.
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:12
$begingroup$
@LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual...
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:13
$begingroup$
@LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual...
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:13
$begingroup$
You need more than that asymptotic.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 19:13
$begingroup$
You need more than that asymptotic.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 19:13
add a comment |
0
active
oldest
votes
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: "69"
};
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%2fmath.stackexchange.com%2fquestions%2f3041664%2fdoes-bounded-version-of-lagranges-4-square-theorem-follow-from-this-result%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- 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%2fmath.stackexchange.com%2fquestions%2f3041664%2fdoes-bounded-version-of-lagranges-4-square-theorem-follow-from-this-result%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
1
$begingroup$
It's not true that $S_4(n)simfrac{pi^2}{2}n^2$. What is true is that $sum_{k=1}^nS_4(k)simfrac{pi^2}{2}n^2$.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 16:16
$begingroup$
Ah yes, I meant that! Let me just update this.
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:12
$begingroup$
@LordSharktheUnknown: Can you answer the question, though? I know from experience on this site that you're quite a capable individual...
$endgroup$
– Isky Mathews
Dec 15 '18 at 17:13
$begingroup$
You need more than that asymptotic.
$endgroup$
– Lord Shark the Unknown
Dec 15 '18 at 19:13