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Exercise 1.1 (Euclid’s Theorem). Show that there are infinitely many prime numbers.

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1 Prime Numbers

A prime number is a positive integer other than 1 that is only divisible by 1 and itself.

As you will show in Exercise 1.1, there are infinitely many primes. The number of primes that are smaller than a given natural number n is denoted π(n).

Exercises

Exercise 1.1 (Euclid’s Theorem). Show that there are infinitely many prime numbers.

Exercise 1.2 . Find an asymptotic formula for π(n). Hint: You might find Exercise 2.1 helpful.

2 Zeta function

The zeta function is given by ζ(s) = P ∞

n=1 n −s , where s is a complex number with real part bigger than 1. For example ζ(2) = π 6

2

.

Exercises

Exercise 2.1 . Extend ζ as far as possible and find all zeros of the function.

1

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