10 1. Arithmetic Functions 1.4. The Mertens estimates Sums p≤x f(p) over primes occur frequently in analytic number theory. In the simpler cases, f is a positive, continuous, monotone function of a real variable that does not change rapidly. If the sum diverges as x +∞, its asymptotic behavior may be guessed from the heuristic p≤x f(p) 2≤n≤x f(n) log(n) x 2 f(u) du log(u) , which is inspired by the observation that the density of the primes near n is close to 1/ log(n). The latter statement is a formulation of the Prime Number Theorem, and indeed the PNT with a good estimate for the error term is a natural tool with which to estimate such sums. As an example of the heuristic in action, consider the sum of log(p)/p over primes p x. The guess for the asymptotic behavior is p≤x log(p) p 2≤n≤x log(n)/n log(n) x 2 du u log(x), and this is actually correct. Indeed, F. C. J. Mertens proved in 1874 that the absolute error in the asymptotic approximation is bounded. This may be expressed as p≤x log(p) p = log(x) + O(1) by means of the big-O notation for the error term. The heuristic for guessing the asymptotic behavior of sums over primes will not perform satisfactorily if f does not have the nice properties assumed. If, for example, f changes sign, there will be cancellation in the sum, and the underlying rationale for the heuristic does not take account of this. Partial summation is perhaps the tool most frequently applied in analytic number theory. The basic version is the identity n m=1 ambm = bn n m=1 am n−1 m=1 (bm+1 bm) m k=1 ak. This is an analogue, for sums, of integration by parts. The partial sum- mation identity is easily proved by observing that bj(a1 + · · · + aj) = bj(a1 + · · · + aj−1)+ ajbj and applying mathematical induction. The partial summation identity yields a formula that is very convenient for estimating weighted sums of arithmetic functions.
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