Help : Deriving an inequality

Andrew99 · 2019-08-06, 01:32

Hi, I am studying a paper by Yann Bugeaud (click here), on page 13 there is an inequality (16) as given below image-

https://www.mersenneforum.org/attach...1&d=1565055006

which is obtained from the below image file -

https://www.mersenneforum.org/attach...1&d=1565055006

, 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-

https://www.mersenneforum.org/attach...1&d=1565055006

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

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

Thanks in Advance.

Uncwilly · 2019-08-06, 03:07

So, you didn't find help over here:
https://www.freemathhelp.com/forum/t...quence.117311/

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.

2019-08-06, 01:32	#1
Andrew99 Aug 2019 11₁₀ Posts	Help : Deriving an inequality Hi, I am studying a paper by Yann Bugeaud (click here), on page 13 there is an inequality (16) as given below image- https://www.mersenneforum.org/attach...1&d=1565055006 which is obtained from the below image file - https://www.mersenneforum.org/attach...1&d=1565055006 , 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- https://www.mersenneforum.org/attach...1&d=1565055006 It is not clear how those two questions would be resolved. Can any one show the derivation of inequality (16)? Thanks in Advance. Attached Thumbnails

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2019-08-06, 03:07	#2
Uncwilly 6809 > 6502 """"""""""""""""""" Aug 2003 101×103 Posts 10001111110101₂ Posts	So, you didn't find help over here: https://www.freemathhelp.com/forum/t...quence.117311/ Since the post is identical except the image attachments it looks like you copied and pasted. Can you give us some more background?

2019-08-06, 04:51	#3
Andrew99 Aug 2019 11 Posts	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.

2019-08-06, 22:51	#4
Dr Sardonicus Feb 2017 Nowhere 2⁴·3²·29 Posts	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.