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HOMEWORK 4 - ELEMENTARY ESTIMATES 1. Assuming the prime number theorem show that for any c > 1 there is an integer x0 (c) such that the interval (x, cx] contains a prime for all x > x0 (c). 2. Recall that X1 p show that X p p diverges. Use the elementary Merten’s results to 1 converges. p log p 3. For an arithmetic function f (n) define f 0 (n) = f (n) log n. Prove the following (i) (f + g)0 = f 0 + g 0 (ii) (f ∗ g)0 = f 0 ∗ g + f ∗ g 0 (iii) I 0 = 0 (iv) (f −1 )0 = −f 0 ∗ (f ∗ f )−1 . (v) U 0 = Λ ∗ U and Λ = U 0 ∗ µ (vi) (Selberg’s Identity) Λ0 + Λ ∗ Λ = U 00 ∗ µ In the following let dn denote the least common multiple of the integers 1, 2, ..., n. The following problems give another proof of Chebyshev’s lower bound. 4. Show that ψ(n) = log dn . 5. Suppose that P (x) isRa polynomial of degree at most n with integer 1 coefficients and I(P ) = 0 P (x)dx. Show that I(P )dn+1 is an integer. 5 6. Show that Q = x2 (1 − x)2 (2x − 1) has |Q(x)| ≤ 5− 2 on [0, 1]. By considering I(Q2k ) deduce that d10k+1 ≥ 55k and hence that ψ(x) ≥ 0.80471 x for all sufficiently large x. 1