Number Theory

Elementary number theory (Math 780) by Filaseta M.

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By Filaseta M.

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To clarify, the last equation in p≤x x = p p≤x x +O p does not assert that a function is O 1 p≤x 1 = p≤x x + O(x) p if and only if it is O(x) but rather there is a p≤x function f (x) that satisfies f (x) = O 1 and f (x) = O(x). Indeed, in the equation p≤x above, the big oh expressions both represent the same function f (x) = p≤x • An estimate using integrals. Explain why k≤x x x − . p p 1 ≥ log x. k Homework: (1) Let f : R+ → R+ and g : R+ → R+ . Find all possible implications between the following.

For example, 11 would be such an integer but 39 would not be. (b) Let A(x) = |{n ≤ x : each of 2, 3, 5, and 7 does not divide n}|. Prove that A(x) ∼ cx for some constant c and determine the value of c. (4) Let a be a real number. Suppose f : [a, ∞) → R has the property that for every t ≥ a, there exists an M (t) such that |f (x)| ≤ M (t) for all x ∈ [a, t]. Suppose g : [a, ∞) → R+ has the property that for every t ≥ a, there exists an ε(t) > 0 such that g(x) ≥ ε(t) for all x ∈ [a, t]. Finally, suppose that f (x) g(x).

Pr is [x/(p1 p2 . . pr )]. The inclusion-exclusion principal implies that the number of positive integers n ≤ x with each prime factor of n being greater than z is [x] − p≤z x + p p1

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