Are Primorials The Worst Case On Euler's Phi Function?
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For each positive integer k, let Pk denote the product of the first k primes. We have φ(Pk)=θ(Pk/loglogPk).
This shows that φ(n) could be as small as (about) n/ log log n for infinitely many n.
Show that this is the “worst case,” in the sense that φ(n) = Ω(n/ log log n).
Can someone please help me? Thanks a lot!
(There is a related question here, but I don't quite understand the explanation. The original questions comes from ntb-v2 Exercise 5.4 and 5.5.)
number-theory asymptotics totient-function
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up vote
-2
down vote
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For each positive integer k, let Pk denote the product of the first k primes. We have φ(Pk)=θ(Pk/loglogPk).
This shows that φ(n) could be as small as (about) n/ log log n for infinitely many n.
Show that this is the “worst case,” in the sense that φ(n) = Ω(n/ log log n).
Can someone please help me? Thanks a lot!
(There is a related question here, but I don't quite understand the explanation. The original questions comes from ntb-v2 Exercise 5.4 and 5.5.)
number-theory asymptotics totient-function
see Nicolas zakuski.utsa.edu/~jagy/Nicolas_1983.pdf
– Will Jagy
Nov 20 at 4:39
also see both my answers at math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:42
proof is in math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:43
add a comment |
up vote
-2
down vote
favorite
up vote
-2
down vote
favorite
For each positive integer k, let Pk denote the product of the first k primes. We have φ(Pk)=θ(Pk/loglogPk).
This shows that φ(n) could be as small as (about) n/ log log n for infinitely many n.
Show that this is the “worst case,” in the sense that φ(n) = Ω(n/ log log n).
Can someone please help me? Thanks a lot!
(There is a related question here, but I don't quite understand the explanation. The original questions comes from ntb-v2 Exercise 5.4 and 5.5.)
number-theory asymptotics totient-function
For each positive integer k, let Pk denote the product of the first k primes. We have φ(Pk)=θ(Pk/loglogPk).
This shows that φ(n) could be as small as (about) n/ log log n for infinitely many n.
Show that this is the “worst case,” in the sense that φ(n) = Ω(n/ log log n).
Can someone please help me? Thanks a lot!
(There is a related question here, but I don't quite understand the explanation. The original questions comes from ntb-v2 Exercise 5.4 and 5.5.)
number-theory asymptotics totient-function
number-theory asymptotics totient-function
asked Nov 20 at 4:36
SamTulster
1
1
see Nicolas zakuski.utsa.edu/~jagy/Nicolas_1983.pdf
– Will Jagy
Nov 20 at 4:39
also see both my answers at math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:42
proof is in math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:43
add a comment |
see Nicolas zakuski.utsa.edu/~jagy/Nicolas_1983.pdf
– Will Jagy
Nov 20 at 4:39
also see both my answers at math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:42
proof is in math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:43
see Nicolas zakuski.utsa.edu/~jagy/Nicolas_1983.pdf
– Will Jagy
Nov 20 at 4:39
see Nicolas zakuski.utsa.edu/~jagy/Nicolas_1983.pdf
– Will Jagy
Nov 20 at 4:39
also see both my answers at math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:42
also see both my answers at math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:42
proof is in math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:43
proof is in math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:43
add a comment |
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see Nicolas zakuski.utsa.edu/~jagy/Nicolas_1983.pdf
– Will Jagy
Nov 20 at 4:39
also see both my answers at math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:42
proof is in math.stackexchange.com/questions/301837/…
– Will Jagy
Nov 20 at 4:43