By Einar Hille

Moment version. This well-known paintings is a textbook that emphasizes the conceptual and old continuity of analytic functionality thought. the second one quantity broadens from a textbook to a textbook-treatise, protecting the ``canonical'' issues (including elliptic capabilities, whole and meromorphic capabilities, in addition to conformal mapping, etc.) and different subject matters closer the increasing frontier of analytic functionality idea. within the latter class are the chapters on majorization and on capabilities holomorphic in a half-plane.

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PRIMES IN ARITHMETIC PROGRESSION: THE GENERAL MODULUS 33 The integral on the right is absolutely convergent for (f > 0, and uniformly for (f ~ {) > 0, and so represents a regular function of s for (f > 0. Thus '(s) is meromorphic fQr (f > 0, its only pole being a simple pole at s = 1 with residue 1. In view of (6), the same is true of L(s, Xo), except that the residue is n (1 - p-I) = q-I 1.

3 4 Landau, Vorlesungen, Satz 202. Landau. Vorlesungen. Satz 111. 46 MULTIPLICATIVE NUMBER THEORY We now turn to the question of the total number of representations of a positive integer n by a representative set of forms of given (fundamental) discriminant d. This question was answered (implicitly, at least) in the classical theory of quadratic forms, developed by Lagrange and further by Gauss. If d < 0, so that the forms are positive definite, the number of representations of n by any form is finite.

The number of primary representations of a given integer n by a given form is finite, since the product of the linear forms x - (Jy and x - (J'y is nfa by (7), and their quotient is bounded both ways by (10). For d > 0 we denote by R(n) the total number of primary representations of n by a representative set of forms of discriminant d. The basic result of the theory of quadratic forms is as follows. 5 , Landau, Voriesullgell, Salz 204. 47 DIRICHLET'S CLASS NUMBER FORMULA lfn > 0 and (n, d) = 1 then R(n) = w (12) L (~), min where w is given by (3) if d < 0 and w = 1 if d > O.