NUMBER-THEORY EXERCISES, VI

DAVID PIERCE

The variables n, k, and d range over the positive integers. Exercise 1. Assuming p is an odd prime:

(a) (p − 1)! ≡ p − 1 (mod 1 + 2 + ⋯ + (p − 1));

(b) 1 · 3 ⋯ (p − 2) ≡ (−1)^{(p−1)/2} · (p − 1) · (p − 3) ⋯ 2 (mod p);

(c) 1 · 3 ⋯ (p − 2) ≡ (−1)^{(p−1)/2} · 2 · 4 ⋯ (p − 1) (mod p);

(d) 1^{2} · 3^{2} ⋯ (p − 2)^{2} ≡ (−1)^{(p+1)/2} (mod p). Exercise 2. τ(n) ⩽ 2\sqrt{n}. Exercise 3. τ(n) is odd if and only if n is square. Exercise 4. Assuming n is odd: σ(n) is odd if and only if n is square.

Exercise 5. ∑_{d|n} \frac{1}{d} = \frac{σ(n)}{n}.

Exercise 6. {n: τ(n) = k} is infinite (when k > 1), but {n: σ(n) = k} is finite. Exercise 7. Let m ∈ ℤ. The number-theoretic function n ↦ n^{m} is multiplicative. Exercise 8. Let ω(n) be the number of distinct prime divisors of n, and let m be a non-zero integer. Then n ↦ m^{ω(n)} is multiplicative.

Exercise 9. Let Λ(n) = \begin{cases}log p, & if n = p^{m} for some positive m; \\ 0, & otherwise. \end{cases}

(a) log n = ∑_{d|n} Λ(d).

(b) Λ(n) = ∑_{d|n} μ(\frac{n}{d}) log d.

(c) Λ(n) = − ∑_{d|n} μ(d) log d.

Exercise 10. Suppose n = p_{1}^{k(1)} ⋯ p_{r}^{k(r)}, where the p_{i} are distinct. (a) If f is multiplicative and non-zero, then ∑_{d|n} μ(d) · f(d) = ∏_{i=1}^{r}(1 −

f(p_{i}));

(b) ∑_{d|n} μ(d) · τ(d) = (−1)^{r}.

Date: November 8, 2007.

Mathematics Dept, Middle East Tech. Univ., Ankara 06531, Turkey E-mail address: dpierce@metu.edu.tr URL: http://www.math.metu.edu.tr/~dpierce/