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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  • 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















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







share|cite|improve this question






















  • 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













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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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


















  • 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















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