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

By Karl-Heinz Fieseler

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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.

5. For an integral domain R with q = |R∗ | < ∞ we have q = lcm {ord(a); a ∈ R∗ } . Proof. The polynomial f = X n − 1 ∈ R[X] with n := lcm {ord(a); a ∈ R∗ } vanishes on R∗ and has at most n zeros, hence q ≤ n. Since by Th. 9 we have n|q it follows q = n. 6. For a non-integral domain the statement does not hold: 1. For R = Z8 we have R∗ = 1, 3, 5, 7 , hence q = 4 and n = 2. 2. Let p, r be two different odd primes. The group of units Z∗pr ∼ = Z∗p × Z∗r has q = (p − 1)(r − 1) elements, but a =1 holds already for = (p − 1)(r − 1)/2, since p − 1 and r − 1.

48,005150881167159727... 49,773832477672302181 52,970321477714460644... 56,446247697063394804... 59,347044002602353079... 60,831778524609809844... 65,112544048081606660... 67,079810529494173714... 69,546401711173979252... 72,067157674481907582... 75,704690699083933168... 77,144840068874805372... 2. There are no zeros on the boundary lines (z) = 0, 1. 3. 3. For a number a ∈ 1 ,1 2 the following statements are equivalent: (a) Z ⊂ [1 − a, a], (b) The Dirichlet series Φµ (z) belonging to the M¨obius µ-function converges in R>a , (c) For all ε > 0 we have π(x) − li(x) = O(x1/2+ε ) The implication ”(b) =⇒ (a)” is easy: We have ζ(z)Φµ (z) = Φ1 (z)Φµ (z) = Φ1∗µ (z) = Φδ (z) = 1 for z ∈ R>1 , hence by analytic continuation ζ(z)Φµ (z) = 1 60 for z ∈ R>a \ {1}, in particular ζ(z) = 0.