A computation in Conrey's paper on Riemann zeta function
I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here).
I have question/doubt in a particular step: In P.10, it claimed that $B=0$. I wonder why it is true, because, in my opinion, there should be an extra term coming from the integration by parts:
$$B=thetaint_0^1 w(y)overline{w}'(y)dy
=thetaint_0^1 e^{2Ry}[R(1+lambda y)^2+lambda(1+lambda y)]dy\
=thetaleft(int_0^1 e^{2Ry}R(1+lambda y)^2dy+frac{1}{2}int_0^1e^{2Ry}d((1+lambda y)^2)right)\
\
=theta Bigg[frac{1}{2}e^{2Ry}(1+lambda y)^2Bigg]_{y=0}^{y=1},$$
where the last equality follows from doing integration by parts. I think this paper have been read and checked by many people. I would appreciate if I can get help from some of you who are familiar with this paper. Thank you very much.
number-theory riemann-zeta zeta-functions riemann-hypothesis
asked Nov 27 at 5:10
Tong
184
add a comment |
I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here).
I have question/doubt in a particular step: In P.10, it claimed that $B=0$. I wonder why it is true, because, in my opinion, there should be an extra term coming from the integration by parts:
$$B=thetaint_0^1 w(y)overline{w}'(y)dy
=thetaint_0^1 e^{2Ry}[R(1+lambda y)^2+lambda(1+lambda y)]dy\
=thetaleft(int_0^1 e^{2Ry}R(1+lambda y)^2dy+frac{1}{2}int_0^1e^{2Ry}d((1+lambda y)^2)right)\
\
=theta Bigg[frac{1}{2}e^{2Ry}(1+lambda y)^2Bigg]_{y=0}^{y=1},$$
where the last equality follows from doing integration by parts. I think this paper have been read and checked by many people. I would appreciate if I can get help from some of you who are familiar with this paper. Thank you very much.
number-theory riemann-zeta zeta-functions riemann-hypothesis
asked Nov 27 at 5:10
Tong
184
I cannot access the paper. The only thing I could say is that your integration is perfect. Now, the stupid question : is there any relation between $R$ and $lambda$ ?
– Claude Leibovici
Nov 27 at 5:37
@ClaudeLeibovici Thanks for the comment. But it seems that $R$ and $lambda$ are not related. In P.10 of the paper, it was chosen such that $R=1.2$ and $lambda=-1.02$, and it does not make the above term vanishes. I am not sure if I miss something.
– Tong
Nov 27 at 11:51
Now posted to MO, mathoverflow.net/questions/316712/…
– Gerry Myerson
Dec 2 at 22:07
add a comment |
I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here).
I have question/doubt in a particular step: In P.10, it claimed that $B=0$. I wonder why it is true, because, in my opinion, there should be an extra term coming from the integration by parts:
$$B=thetaint_0^1 w(y)overline{w}'(y)dy
=thetaint_0^1 e^{2Ry}[R(1+lambda y)^2+lambda(1+lambda y)]dy\
=thetaleft(int_0^1 e^{2Ry}R(1+lambda y)^2dy+frac{1}{2}int_0^1e^{2Ry}d((1+lambda y)^2)right)\
\
=theta Bigg[frac{1}{2}e^{2Ry}(1+lambda y)^2Bigg]_{y=0}^{y=1},$$
where the last equality follows from doing integration by parts. I think this paper have been read and checked by many people. I would appreciate if I can get help from some of you who are familiar with this paper. Thank you very much.
number-theory riemann-zeta zeta-functions riemann-hypothesis
asked Nov 27 at 5:10
Tong
184
I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here).
I have question/doubt in a particular step: In P.10, it claimed that $B=0$. I wonder why it is true, because, in my opinion, there should be an extra term coming from the integration by parts:
$$B=thetaint_0^1 w(y)overline{w}'(y)dy
=thetaint_0^1 e^{2Ry}[R(1+lambda y)^2+lambda(1+lambda y)]dy\
=thetaleft(int_0^1 e^{2Ry}R(1+lambda y)^2dy+frac{1}{2}int_0^1e^{2Ry}d((1+lambda y)^2)right)\
\
=theta Bigg[frac{1}{2}e^{2Ry}(1+lambda y)^2Bigg]_{y=0}^{y=1},$$
where the last equality follows from doing integration by parts. I think this paper have been read and checked by many people. I would appreciate if I can get help from some of you who are familiar with this paper. Thank you very much.
number-theory riemann-zeta zeta-functions riemann-hypothesis
number-theory riemann-zeta zeta-functions riemann-hypothesis
asked Nov 27 at 5:10
Tong
184
asked Nov 27 at 5:10
Tong
184
asked Nov 27 at 5:10
Tong
184
asked Nov 27 at 5:10
Tong
184
asked Nov 27 at 5:10
Tong
184
184
I cannot access the paper. The only thing I could say is that your integration is perfect. Now, the stupid question : is there any relation between $R$ and $lambda$ ?
– Claude Leibovici
Nov 27 at 5:37
@ClaudeLeibovici Thanks for the comment. But it seems that $R$ and $lambda$ are not related. In P.10 of the paper, it was chosen such that $R=1.2$ and $lambda=-1.02$, and it does not make the above term vanishes. I am not sure if I miss something.
– Tong
Nov 27 at 11:51
Now posted to MO, mathoverflow.net/questions/316712/…
– Gerry Myerson
Dec 2 at 22:07
add a comment |
I cannot access the paper. The only thing I could say is that your integration is perfect. Now, the stupid question : is there any relation between $R$ and $lambda$ ?
– Claude Leibovici
Nov 27 at 5:37
@ClaudeLeibovici Thanks for the comment. But it seems that $R$ and $lambda$ are not related. In P.10 of the paper, it was chosen such that $R=1.2$ and $lambda=-1.02$, and it does not make the above term vanishes. I am not sure if I miss something.
– Tong
Nov 27 at 11:51
Now posted to MO, mathoverflow.net/questions/316712/…
– Gerry Myerson
Dec 2 at 22:07
I cannot access the paper. The only thing I could say is that your integration is perfect. Now, the stupid question : is there any relation between $R$ and $lambda$ ?
– Claude Leibovici
Nov 27 at 5:37
I cannot access the paper. The only thing I could say is that your integration is perfect. Now, the stupid question : is there any relation between $R$ and $lambda$ ?
– Claude Leibovici
Nov 27 at 5:37
@ClaudeLeibovici Thanks for the comment. But it seems that $R$ and $lambda$ are not related. In P.10 of the paper, it was chosen such that $R=1.2$ and $lambda=-1.02$, and it does not make the above term vanishes. I am not sure if I miss something.
– Tong
Nov 27 at 11:51
@ClaudeLeibovici Thanks for the comment. But it seems that $R$ and $lambda$ are not related. In P.10 of the paper, it was chosen such that $R=1.2$ and $lambda=-1.02$, and it does not make the above term vanishes. I am not sure if I miss something.
– Tong
Nov 27 at 11:51
Now posted to MO, mathoverflow.net/questions/316712/…
– Gerry Myerson
Dec 2 at 22:07
Now posted to MO, mathoverflow.net/questions/316712/…
– Gerry Myerson
Dec 2 at 22:07
add a comment |
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%2f3015359%2fa-computation-in-conreys-paper-on-riemann-zeta-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3015359%2fa-computation-in-conreys-paper-on-riemann-zeta-function%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
I cannot access the paper. The only thing I could say is that your integration is perfect. Now, the stupid question : is there any relation between $R$ and $lambda$ ?
– Claude Leibovici
Nov 27 at 5:37
@ClaudeLeibovici Thanks for the comment. But it seems that $R$ and $lambda$ are not related. In P.10 of the paper, it was chosen such that $R=1.2$ and $lambda=-1.02$, and it does not make the above term vanishes. I am not sure if I miss something.
– Tong
Nov 27 at 11:51
Now posted to MO, mathoverflow.net/questions/316712/…
– Gerry Myerson
Dec 2 at 22:07