Devoir libre n 7

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n≥2n
∀a∈N∗, a ∧n= 1 =⇒an−1≡1 [n].
n≥2r≥2
p1, ..., pr
n=p1...pr
∀i∈J1, rKpi−1|n−1
a∈N∗a∧n= 1 i∈J1, rKan−1≡1 [pi]
n
561 = 3 ×11 ×17
m≥2k∈Zl≥1kl≡1 [m]
r∈N∗i∈N
ki≡1 [m]⇐⇒ r|i.
r k m
n≥2n
p, q ∈N∗p n =p2q a = 1 + pq
an−1≡1 [n] (1 + pq)(1 −pq)
p a n
f:N∗→Rf
∀(n, m)∈(N∗)2, f(nm) = f(n)f(m)
∀(n, m)∈(N∗)2, n ∧m=1=⇒f(nm) = f(n)f(m)
n∈N∗r(n) = |{p∈ P | νp(n)6= 1}|
n
µ(n) = (−1)r(n)∀p∈ P, νp(n)≤1
0
µ
f:N∗→R
f(1)
n∈N∗f(n)>0
g= ln(f) (n, m)∈(N∗)2
n∧m=1=⇒g(nm) = g(n) + g(m).
a∈N∗a≥2
k∈Na∧(ak+... + 1) = 1
k∈N∗
g(ak−(ak−1+... + 1)) ≤kg(a)≤g(ak+ak−1+... + 1).
a∈N∗a≥2k∈N∗
jk
3jk−1−(3jk−2+... + 1) ≤ak+... + 1 <3jk−(3jk−1+... + 1).
jka k
kg(a)≤jkg(3)
jk
jk
k−−−−−→
k→+∞
ln(a)
ln(3) .
g(a)≤ln(a)
ln(3) g(3).

a∈N∗a≥3k∈N∗
mk∈N
3mk+... + 3 + 1 ≤ak−(ak−1+... + 1) <3mk+1 +... + 3 + 1.
mka k
mk
k−−−−−→
k→+∞
ln(a)
ln(3) .
g(a)≥ln(a)
ln(3) g(3).
α∈R∗
∀n∈N∗, n ≥3 =⇒g(n) = αln(n).
n= 1 n= 2 g(6)
n∈N∗f(n) = nα