# Euler's totient function and Riemann hypothesis

I am looking for an upper-bound of the Euler's totient function $$\varphi$$ which would be equivalent to the Riemann hypothesis (RH). There is the following Nicolas' criterion about primorial numbers $$N_k$$ (product of the $$k$$ first prime numbers):

Nicolas' criterion (MR724536, Théorème 2)

• If RH is true, then for all $$k \ge 1$$ $$N_k / \varphi(N_k) > e^{\gamma}\ln \ln N_k$$
• If RH is false then there are infinitely many $$k$$ such that the above inequality holds, and infinitely many $$k$$ such that it does not.

In particular, RH is equivalent to the following upper-bound: $$\varphi(n)< \frac{n}{e^{\gamma}\ln \ln n}, \text{ for } n=N_k \text{ and } k>1.$$

Now, the above inequality does not hold for $$n$$ prime greater than $$7$$. So an upper-bound for all $$n$$ requires a modification. Let $$\omega(n)$$ be the number of distinct prime factors of $$n$$. Then $$N_k$$ is the smallest number $$n$$ satisfying $$\omega(n)=k$$. Consider the following step function $$s=\sum_{k=1}^{\infty}k\chi_{(N_{k-1},N_k]}.$$

Question: Is RH equivalent to the following upper-bound? $$\varphi(n)< \frac{n2^{s(n)-\omega(n)}}{e^{\gamma}\ln \ln n}, \text{ for } n>2.$$

Because $$s(n) = \omega(n)$$ iff $$n$$ is primorial (i.e. of the form $$N_k$$), RH is equivalent to this upper-bound for $$n$$ primorial, by Nicolas' criterion. So it remains to deduce (from RH) this upper-bound for $$n$$ non-primorial. Consider the map $$\alpha$$ defined by $$\alpha(N_k)=1$$ and for $$n$$ non-primorial by $$\alpha(n)=\left( \frac{\varphi(n)e^{\gamma}\ln \ln n}{n} \right)^{1/(s(n)-\omega(n))}$$ Then it remains to deduce that $$\alpha(n)<2$$.

Proposition: There is a sequence $$(n_r)$$ such that $$\alpha(n_r) \to 2$$.

Proof: Let $$p_r$$ be the r-th prime number. Let $$q_r$$ be the prime number just before $$2p_r$$ (see A059788), by Bertrand–Chebyshev theorem, $$q_r \neq p_r$$.
Now consider the number $$n_r = N_rq_r/(2p_r)$$. Then $$s(n_r)-\omega(n_r)=1$$. So $$\alpha(n_r) = \frac{\varphi(n_r)e^{\gamma}\ln \ln n_r}{n_r} = \frac{\varphi(N_r)e^{\gamma}\ln \ln N_r}{N_r} \cdot \frac{2(q_r-1)p_r \ln \ln (n_r)}{(p_r-1)q_r\ln \ln (N_r)}.$$ The first component of this multiplication converges to $$1_-$$, by Rosser-Schoenfeld and Nicolas's criterion (assuming RH). Then $$\alpha(n_r) \sim \frac{2(q_r-1)p_r \ln \ln (n_r)}{(p_r-1)q_r\ln \ln (N_r)} = 2 \cdot \frac{p_rq_r-p_r}{p_rq_r-q_r} \cdot \frac{\ln ( \ln (N_r) - \ln (2p_r/q_r))}{\ln \ln N_r} \to 2. \ \ \square$$ It follows that if the expected upper-bound is true, then it is optimal.

A number $$N>2$$ is called a champion if $$\alpha(N)>1$$ and $$\forall n we have $$\alpha(n)<\alpha(N)$$.
There are exactly $$35$$ champions $$N< 10^{10}$$, listed below together with $$\alpha(N)$$ and the factor decomposition of the ratio with their next primorial.

sage: champions(3,10000000000)
[7, 1.0081297159194946, 2 * 3 * 5 * 7^-1]
[11, 1.1900001764297485, 2 * 3 * 5 * 11^-1]
[13, 1.244431734085083, 2 * 3 * 5 * 13^-1]
[17, 1.3212575912475586, 2 * 3 * 5 * 17^-1]
[19, 1.349881649017334, 2 * 3 * 5 * 19^-1]
[23, 1.395310401916504, 2 * 3 * 5 * 23^-1]
[29, 1.4449400901794434, 2 * 3 * 5 * 29^-1]
[91, 1.4570322036743164, 2 * 3 * 5 * 13^-1]
[119, 1.499191164970398, 2 * 3 * 5 * 17^-1]
[133, 1.5151351690292358, 2 * 3 * 5 * 19^-1]
[143, 1.5473616123199463, 2 * 3 * 5 * 7 * 11^-1 * 13^-1]
[187, 1.5879234075546265, 2 * 3 * 5 * 7 * 11^-1 * 17^-1]
[209, 1.6032350063323975, 2 * 3 * 5 * 7 * 11^-1 * 19^-1]
[1309, 1.6044903993606567, 2 * 3 * 5 * 17^-1]
[1463, 1.61602783203125, 2 * 3 * 5 * 19^-1]
[1547, 1.6261959075927734, 2 * 3 * 5 * 11 * 13^-1 * 17^-1]
[1729, 1.6376748085021973, 2 * 3 * 5 * 11 * 13^-1 * 19^-1]
[2093, 1.6558983325958252, 2 * 3 * 5 * 11 * 13^-1 * 23^-1]
[2261, 1.6681385040283203, 2 * 3 * 5 * 11 * 17^-1 * 19^-1]
[23023, 1.6813620328903198, 2 * 3 * 5 * 23^-1]
[24871, 1.692425012588501, 2 * 3 * 5 * 13 * 17^-1 * 19^-1]
[29029, 1.6975822448730469, 2 * 3 * 5 * 29^-1]
[29393, 1.7114139795303345, 2 * 3 * 5 * 11 * 17^-1 * 19^-1]
[391391, 1.7167580127716064, 2 * 3 * 5 * 23^-1]
[437437, 1.7252918481826782, 2 * 3 * 5 * 17 * 19^-1 * 23^-1]
[493493, 1.7308218479156494, 2 * 3 * 5 * 29^-1]
[7436429, 1.7369874715805054, 2 * 3 * 5 * 23^-1]
[8580495, 1.74891197681427, 2 * 13 * 23^-1]
[9376367, 1.7497258186340332, 2 * 3 * 5 * 29^-1]
[181996815, 1.7533072233200073, 2 * 19 * 31^-1]
[190285095, 1.7621831893920898, 2 * 17 * 29^-1]
[203408205, 1.7683358192443848, 2 * 17 * 31^-1]
[5203883685, 1.7786405086517334, 2 * 23 * 37^-1]
[5277907635, 1.786533236503601, 2 * 19 * 31^-1]
[5898837945, 1.8011537790298462, 2 * 17 * 31^-1]


Note that all these champions are closely related to their next primorial numbers.

Then, the expected upper-bound is checked for $$n<10^{10}$$.
If it is true in general then it must have infinitely many champions, forced by the sequence $$(n_r)$$.
See below $$[n_r,\alpha(n_r)]$$ for $$r=11,12$$:

[197325643515, 1.8032277291323942]
[7320457889745, 1.8197057337162745]


Code

# %attach SAGE/EulerRH.spyx

from sage.all import *

cpdef g(float x):
return x/(exp(float(euler_gamma))*ln(ln(x)))

cpdef omega(long n):
return len(list(factor(n)))

cpdef champions(long m1, long m2):
cdef long n,o,s,p,pr
cdef float a,c
a=1; s=2; p=3; pr=6; n=m1
while pr<m1:
p=next_prime(p); pr*=p; s+=1
while n<=m2:
#if mod(n,10**7)==0:
#print(n)
if n>pr:
p=next_prime(p); pr*=p; s+=1
o=omega(n)
if n<>pr:
c=(euler_phi(n)/g(float(n)))**(1/float(s-o))
if c>a:
a=c
print([n,a,factor(Integer(pr)/Integer(n))])
n+=1