Elementary Number Theory: Some Lecture Notes by Karl-Heinz Fieseler

By Karl-Heinz Fieseler

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Before we discuss how to find an expression for ζ outside the right half plane R>1 , we note that ζ is unique if it exists and that ζ(z) = 0 for all z ∈ R>1 . 1) and the fact that the locally uniform limit f : G −→ C of a sequence of nowhere vanishing holomorphic functions fn : G −→ C on a domain G is either f ≡ 0 or has itself no zeros. ) For every n ∈ N there is an expression of the first type on the right half plane R>−n . We discuss only the case n = 0. 1 z −z Φ1 (z) = z−1 ∞ 1 with the Gauß bracket [t] = max Z≤t and tz := ez ln t .

Finally we arrive at an infinite p-adic expansion ν+1 ≡ ν mod (p ∞ tn pn a= n=0 with unique digits tn , 0 ≤ tn < p. 31 6. In order to expand a real number we can also work with the prime p as basis (instead of 10), but get series in the opposite direction ∞ R tn p−n . x= n= 7. Let us return to p-adic integers: We have 1 − p ∈ Z∗(p) and ∞ −1 (1 − p) pn , = n=0 so a negative rational number is the infinite sum of positive numbers: This shows that p-adic integers do not admit an order relation compatible with ring operations.

2. Furthermore −1 p = (−1) p−1 2 1 , if p ≡ 1 −1 , if p ≡ 3 = mod (4) mod (4) . 3. The law of quadratic reciprocity: For a prime q ∈ P>2 different from p we have p−1 q−1 q p = (−1) 2 2 . p q With other words q p =− 43 p q if p ≡ q ≡ 3 mod (4), and q p = p q otherwise. 5. We compute 42 61 = (−1) · = 1 3 2 61 3 61 −2 7 =− 7 61 −1 7 = (−1) · 2 7 61 3 61 7 = −(−1) · 1 = 1. 4 we take for granted the existence of a field K ⊃ Zp , such that there is an element ζ ∈ K ∗ of order 8 resp. an element η ∈ K ∗ of order q.

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