• Aucun résultat trouvé

If f is a multiplicative function, prove that X d|n µ(d)f(d

N/A
N/A
Protected

Academic year: 2022

Partager "If f is a multiplicative function, prove that X d|n µ(d)f(d"

Copied!
1
0
0

Texte intégral

(1)

HOMEWORK VI

MATH-UA 0248-001 THEORY OF NUMBERS due on October, 27, 2017

1. Prove that there are infinitely many pairs of integers m and n with σ(m2) = σ(n2). (Hint: look atm = 5k, n= 4k with (k,10) = 1).

2. Show that if 2k−1 is a prime, then n = 2k−1(2k−1) satisfies the equation σ(n) = 2n.

3. For each positive integer n show that

µ(n)µ(n+ 1)µ(n+ 2)µ(n+ 3) = 0.

4. Let n =pα11pα22. . . pαrr be the prime factorization of the integern >1. If f is a multiplicative function, prove that

X

d|n

µ(d)f(d) = (1−f(p1))(1−f(p2)). . .(1−f(pr)).

(Hint: is the function F(n) = P

d|nµ(d)f(d) multiplicative?)

5. Let n =pα11pα22. . . pαrr be the prime factorization of the integer n >1. Prove that

(a) P

d|nµ(d)σ(d) = (−1)rp1p2. . . pr; (b) P

d|nµ(d)/d= (1− p1

1)(1− p1

2). . .(1− p1

r).

1

Références

Documents relatifs

Indiquer, pour le président du club, la formule la plus avantageuse pour un voyage de dix personnes.. Justifier la réponse par

Indiquer, pour le président du club, la formule la plus avantageuse pour un voyage de dix personnes.. Justifier la réponse par

Déterminer l'encadrement de la masse d’engrais azoté répandu pour que l’exploitant agricole ne soit pas pénalisé.a. Lors d’un lancer franc au basket, le joueur

In this section we give some elementary inequalities for affine maps in Ro and consider the approximation of QS maps by similarities.. We first introduce

is an open hyperbolic triangular region qrith a vertex angle at least a at the vertex zn, and the hyperbolic distance from zn to the opposite side of the

of convex D and thin ring-shaped R, but before explaining in detail exactly which regions we consider we motivate what is to follow with some heuristic

The notion of a weakly quasisymmetric embed- ding was introduced by Tukia and Våiisälä [TV]... Thefunction n*u(n')

The second property of the capacity metric that we derive is an interpretation of this metric in terms of the reduced modulus of a path family of cycles homologous