Inlämningsuppgift 1

1.2 Find integers x, y, z such that 35x + 55y + 77z = 1. 1.3 Prove that

∑_{i=1}^{n} \frac{1}{i}

is not an integer for any n > 1. 1.4 Prove that

({a, b}, {b, c}, {c, a}) = {(a, b), (b, c), (c, a)} ,

where (x, y) and {x, y} denote, respectively, the greatest common divisor and least common multiple of the numbers x and y. 1.8 Show that, if m > n, then 2^{2^{n}} + 1 divides 2^{2^{m}} − 1, and so (2^{2^{m}} + 1, 2^{2^{n}} + 1) = 1. 1.9 Deduce from 1.8 that p_{n+1} ≤ 2^{2^{n}} + 1, whence π(x) ≥ log log x, for all x ≥ 2. 2.2 Let Λ(n) = log p if n is a power of a prime p, and let Λ(n) = 0 for all n which are not prime powers. Evaluate

∑_{d|n} Λ(d).

Express

∑_{n=1}^{∞} \frac{Λ(n)}{n^{s}}

in terms of ζ(s). 2.4 Let n > 0. Let a run through the integers from 1 to n such that (a, n) = 1. Prove that

\frac{1}{n^{3}} ∑ a^{3} = \frac{1}{4}ϕ(n) (1 + (−1)^{k}p_{1} ⋯ p_{k}/n^{2}) ,

where p_{1}, ..., p_{k} are the distinct prime factors of n. 2.6 Show that, for any real nbumber x ≥ 1,

∑_{n≤x} μ(n)⌊x/n⌋ = 1.

Hence prove that

|∑_{n≤x} \frac{μ(n)}{n}| ≤ 1.

2.8 Prove that

∑_{n=1}^{∞} ϕ(n)\frac{x^{n}}{1 − x^{n}} = \frac{x}{(1 − x)^{2}} .