36 1. Arithmetic Functions (21) a) Use integration by parts to show that n+1 n f(u) du = f(n) 2 + f(n + 1) 2 − n+1 n u − n − 1 2 f (u) du. Here n is an integer and f a continuous function on the interval [n, n+1] with f piecewise continuous there. b) Use part a) to establish the Euler-Maclaurin summation formula. (22) Prove the estimate ∞ n=−∞ e−πn2u = 1 √ u + O(1) as u → 0+. This series comes from the theory of elliptic theta functions, and satisfies a functional equation that yields a far more precise estimate. (23) † Establish the estimate n≤x logm x n = m!x + O(m!logm(x)) uniformly in positive integers m. (24) Show without integration that T(n) = n log(n) + O(n) by subdividing the interval [1,n] between successive powers of 2. (25) Show that p≤x 1 + 1 p ∼ c log(x) as x → +∞, for some positive constant c. (26) Show that p≤x log2(p) p = 1 2 log2(x) + O(log(x)) as x → +∞. (27) Show that the real number 1 is a point of accumulation of the sequence of ratios (pk+1/pk)k=1 ∞ of successive primes. (28) Show that the series p 1 p(log log(p))2 converges. (29) A divisor d of an integer n is called a block divisor if it is coprime to its complementary divisor n/d. In group theory such divisors are called Hall divisors. Count the number of block divisors of n. (30) Show that the multiplicative arithmetic functions under Dirichlet con- volution constitute a subgroup of the group of all arithmetic functions that have a Dirichlet inverse.
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