xviii Introduction n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 d(n) 1 2 2 3 2 4 2 4 3 4 2 6 2 4 4 Table 1. Values of the divisor function Counting the number of divisors of positive integers leads to a diﬃcult problem known as the Dirichlet Divisor Problem that is still unsolved today. Denoting the number of divisors of n by d(n), Table 1 shows that these counts fluctuate a good deal. But much more regular behavior is revealed by averaging d(n), and in fact 1 x n≤x d(n) ≈ log(x) + 2γ − 1, γ = 0.5772 . . ., with an absolute error that tends to zero as x → +∞. To determine how fast the error tends to zero is the divisor problem of Dirichlet. Divisors and primes are the stuff of multiplicative number theory. But there are also interesting counting problems connected with additive ques- tions. The eighteenth-century English algebraist Edward Waring stated that every positive integer may be expressed as a sum of a limited number of k-th powers of nonnegative integers, the number required depending on k only. We shall count the number of such representations for large integers when the number of powers allowed is suﬃciently large, finding an asymptotic for- mula by means of the Circle Method and establishing Waring’s claim. This was first achieved by David Hilbert by a method different from the one used here. The proof is the most elaborate in the book, though the prerequisites are surprisingly modest. The Circle Method is related to Fourier theory, but involves only Fourier series with finitely many terms so convergence issues do not arise. The achievements of analytic number theory are not entirely limited to approximate counts. Some of the quantities estimated are not counting numbers, and for a few problems exact rather than approximate results have been attained. We shall cover one such case from algebraic number theory, that of the analytic class number formula hK = wK|dK|1/2 2r1(K)+r2(K)πr2(K)RK lim s→1 ζK(s) ζ(s) that expresses the number hK of ideal classes of the ring of algebraic integers of a number field K in terms of other arithmetic data. This formula is due to Dirichlet and Richard Dedekind.

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