Elementary Number Theory (Math 780 instructors notes)

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W for k is incorrect, B veri es that A is incorrect by giving A the factorization of n. 32 De nitions and Notations for Analytic Estimates: Let f and g be real-valued functions with domain containing an interval c; 1) for some real number c. We say that f (x) is big oh of g(x) and write f (x) = O(g(x)) if there is a constant C > 0 such that jf (x)j Cg(x) for all x su ciently large. We say f (x) is less than less than g(x) and write f (x) g(x) if f (x) = O(g(x)), and we say f (x) is greater than greater than g(x) and write f (x) g(x) if g(x) = O(f (x)).

2 m pj pj . Therefore, log p: It is easy to verify that 2x] ? 2 x] 2 f0; 1g for every real number x. Hence, ( ) and ( ) imply m log 2 X p 2m 1 j log(2m)= log p Thus, if n = 2m, then Also, if n = 2m + 1, then (n) X (n) 1 log p X p 2m log(2m) = (2m) log(2m): log 2 n > 1 n : 2 log n 4 log n 2m (2m) > 14 log(2 m) 1 2m 2m + 1 4 2m + 1 log(2m + 1) 1 n : 6 log n 40 This establishes the lower bound in the theorem (for all positive integers n). For the upper bound, we use that if m < p 2m, then 2m=p] ? 2 m=p] = 1.

For every k > 0, we have (x) = Li(x) + O x where the logk x implied constant depends on k. Theorem 35 implies Theorem 34 and more. Using integration by parts and the estimate x Z () 2 dt log4 t x ; log4 x explain why Theorem 35 implies (x) = logx x + x2 + 2x3 + O x4 : log x log x log x Dirichlet's Theorem asserts that if a and b are positive relatively prime integers, then there are in nitely many primes of the form a + bn. Set (x; b; a) = jfp x : p a (mod b)gj: Then a strong variation of Dirichlet's Theorem is the following.

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