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-   -   Help : Deriving an inequality (https://www.mersenneforum.org/showthread.php?t=24666)

 Andrew99 2019-08-06 01:32

Help : Deriving an inequality

3 Attachment(s)
Hi, I am studying a paper by Yann Bugeaud ([URL="http://irma.math.unistra.fr/~bugeaud/travaux/ConfMumbaidef.pdf"]click here[/URL]), on page 13 there is an inequality (16) as given below image-

[IMG]https://www.mersenneforum.org/attachment.php?attachmentid=20857&stc=1&d=1565055006[/IMG]

which is obtained from the below image file -

[IMG]https://www.mersenneforum.org/attachment.php?attachmentid=20858&stc=1&d=1565055006[/IMG]

, on page 12.

How the inequality (16) is derived? I couldn't figure it out. However one of my forum member tried but it has two problems (problems are marked as "how?"), it is given in below image-

[IMG]https://www.mersenneforum.org/attachment.php?attachmentid=20859&stc=1&d=1565055006[/IMG]

It is not clear how those two questions would be resolved.

Can any one show the derivation of inequality (16)?

 Uncwilly 2019-08-06 03:07

So, you didn't find help over here:

Since the post is identical except the image attachments it looks like you copied and pasted. Can you give us some more background?

 Andrew99 2019-08-06 04:51

No I did not get any help from there, you can find all detail in page 12 an 13 of the paper, not much detail is given.

 Dr Sardonicus 2019-08-06 22:51

The symbol $$\ll$$ isn't just an inequality symbol. In analytic number theory, if f and g are functions of one variable (in particular, a positive integer variable)

$$f(x)\;\ll\;g(x)$$

indicates that there is a (positive) constant k such that

$$f(n) \;\le\; k\cdot g(n)$$

for sufficiently large n. The notation is due to Vinogradov.

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