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Number theory exercises WS 2019, TU Graz

Sheet 2, solutions (on paper) to be handed in on 22th October 2019

2-1. Prove: If an+ 1 is prime, then n = 2k, where k ≥ 0 is an integer. (Hint: compare with: if 2n−1 is a prime, then n is a prime.)

2-2. Prove that

gcd(am−1, an−1) = agcd(m,n)−1.

2-3. If p | 22n + 1, then p ≡ 1 mod 2n+1. Conclude that for all n ≥ 2 there exist infinitely many primes p≡1 mod 2n,

2-4. Let f(s) = P∞ n=1

an

ns be a so called Dirichlet series. Here an ∈ C are coefficients and s is a complex variable. Let ζ(s) = P∞

n=1 1

ns. The series is convergent when Re(s)>1. Prove that (ζ(s))2 =

∞

X

n=1

τ(n) ns . 2-5. Prove that P

n≤xτ(n) = xlogx+ (2γ−1)x+O(√

x). (Counting of lattice points.) You can use for example Apostol, Introduction to analytic number theory, Theorem 3.3.

(NOT for crosses, and not to be handed in.) Deadline for crosses are: Tuesday 9.55am.

https://www.math.tugraz.at/∼elsholtz/WWW/lectures/ws19/numbertheory/vorlesung.html

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