“Trivial” assymptotic bound in Gallagher's paper
$begingroup$
In his paper On the distribution of primes in short intervals, right before equation 9, Gallagher states that $$sum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llsum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}frac{d}{x}{log}^B x$$ where C is a fixed constant, $omega(n)$ is the number of prime factors of $n$, $mu$ is the Mobius function and D is given by $$D = sum_{ilt j}(d_i-d_j)$$ in which $1leq d_1lt ... lt d_r leq h$, h being a fixed positive constant and the $d_i$'s are integers.
To my understanding the notation $ll$ means : does not grow faster than, with respect to the variable $x$. The paper can be found here : cambridge.org
What I can't understand is how one gets $$sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llfrac{d}{x}{log}^B x$$
number-theory upper-lower-bounds
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
$endgroup$
add a comment |
$begingroup$
In his paper On the distribution of primes in short intervals, right before equation 9, Gallagher states that $$sum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llsum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}frac{d}{x}{log}^B x$$ where C is a fixed constant, $omega(n)$ is the number of prime factors of $n$, $mu$ is the Mobius function and D is given by $$D = sum_{ilt j}(d_i-d_j)$$ in which $1leq d_1lt ... lt d_r leq h$, h being a fixed positive constant and the $d_i$'s are integers.
To my understanding the notation $ll$ means : does not grow faster than, with respect to the variable $x$. The paper can be found here : cambridge.org
What I can't understand is how one gets $$sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llfrac{d}{x}{log}^B x$$
number-theory upper-lower-bounds
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
$endgroup$
$begingroup$
Unless we have access to the paper, to which you have only given us a partial citation, we cannot help you with this.
$endgroup$
– Lord Shark the Unknown
Jan 6 at 14:03
$begingroup$
A standard method is Selberg-Delange method. This begins with writing $$sum_{n=1}^{infty} frac{mu(n) C^{w(n)}}{phi(n)^2 n^s}$$ as a product of a power of the Riemann zeta function and a function whose abscissa of absolute convergence is $sigma>1-delta$ for some $delta>0$.
$endgroup$
– i707107
Jan 8 at 1:14
add a comment |
$begingroup$
In his paper On the distribution of primes in short intervals, right before equation 9, Gallagher states that $$sum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llsum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}frac{d}{x}{log}^B x$$ where C is a fixed constant, $omega(n)$ is the number of prime factors of $n$, $mu$ is the Mobius function and D is given by $$D = sum_{ilt j}(d_i-d_j)$$ in which $1leq d_1lt ... lt d_r leq h$, h being a fixed positive constant and the $d_i$'s are integers.
To my understanding the notation $ll$ means : does not grow faster than, with respect to the variable $x$. The paper can be found here : cambridge.org
What I can't understand is how one gets $$sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llfrac{d}{x}{log}^B x$$
number-theory upper-lower-bounds
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
$endgroup$
In his paper On the distribution of primes in short intervals, right before equation 9, Gallagher states that $$sum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llsum_{d|D}frac{mu^2(d)*C^{omega(d)}}{phi(d)}frac{d}{x}{log}^B x$$ where C is a fixed constant, $omega(n)$ is the number of prime factors of $n$, $mu$ is the Mobius function and D is given by $$D = sum_{ilt j}(d_i-d_j)$$ in which $1leq d_1lt ... lt d_r leq h$, h being a fixed positive constant and the $d_i$'s are integers.
To my understanding the notation $ll$ means : does not grow faster than, with respect to the variable $x$. The paper can be found here : cambridge.org
What I can't understand is how one gets $$sum_{substack{egt x/d \ text{(e,D)=1}}}frac{mu^2(e)*C^{omega(e)}}{phi^2(e)}llfrac{d}{x}{log}^B x$$
number-theory upper-lower-bounds
number-theory upper-lower-bounds
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
edited Jan 13 at 11:28
TheDabol51
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
asked Jan 6 at 13:57
TheDabol51TheDabol51
12
12
$begingroup$
Unless we have access to the paper, to which you have only given us a partial citation, we cannot help you with this.
$endgroup$
– Lord Shark the Unknown
Jan 6 at 14:03
$begingroup$
A standard method is Selberg-Delange method. This begins with writing $$sum_{n=1}^{infty} frac{mu(n) C^{w(n)}}{phi(n)^2 n^s}$$ as a product of a power of the Riemann zeta function and a function whose abscissa of absolute convergence is $sigma>1-delta$ for some $delta>0$.
$endgroup$
– i707107
Jan 8 at 1:14
add a comment |
$begingroup$
Unless we have access to the paper, to which you have only given us a partial citation, we cannot help you with this.
$endgroup$
– Lord Shark the Unknown
Jan 6 at 14:03
$begingroup$
A standard method is Selberg-Delange method. This begins with writing $$sum_{n=1}^{infty} frac{mu(n) C^{w(n)}}{phi(n)^2 n^s}$$ as a product of a power of the Riemann zeta function and a function whose abscissa of absolute convergence is $sigma>1-delta$ for some $delta>0$.
$endgroup$
– i707107
Jan 8 at 1:14
$begingroup$
Unless we have access to the paper, to which you have only given us a partial citation, we cannot help you with this.
$endgroup$
– Lord Shark the Unknown
Jan 6 at 14:03
$begingroup$
Unless we have access to the paper, to which you have only given us a partial citation, we cannot help you with this.
$endgroup$
– Lord Shark the Unknown
Jan 6 at 14:03
$begingroup$
A standard method is Selberg-Delange method. This begins with writing $$sum_{n=1}^{infty} frac{mu(n) C^{w(n)}}{phi(n)^2 n^s}$$ as a product of a power of the Riemann zeta function and a function whose abscissa of absolute convergence is $sigma>1-delta$ for some $delta>0$.
$endgroup$
– i707107
Jan 8 at 1:14
$begingroup$
A standard method is Selberg-Delange method. This begins with writing $$sum_{n=1}^{infty} frac{mu(n) C^{w(n)}}{phi(n)^2 n^s}$$ as a product of a power of the Riemann zeta function and a function whose abscissa of absolute convergence is $sigma>1-delta$ for some $delta>0$.
$endgroup$
– i707107
Jan 8 at 1:14
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%2f3063890%2ftrivial-assymptotic-bound-in-gallaghers-paper%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%2f3063890%2ftrivial-assymptotic-bound-in-gallaghers-paper%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
$begingroup$
Unless we have access to the paper, to which you have only given us a partial citation, we cannot help you with this.
$endgroup$
– Lord Shark the Unknown
Jan 6 at 14:03
$begingroup$
A standard method is Selberg-Delange method. This begins with writing $$sum_{n=1}^{infty} frac{mu(n) C^{w(n)}}{phi(n)^2 n^s}$$ as a product of a power of the Riemann zeta function and a function whose abscissa of absolute convergence is $sigma>1-delta$ for some $delta>0$.
$endgroup$
– i707107
Jan 8 at 1:14