By P. T. Bateman, Harold G. Diamond

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**Example text**

5) n

There exist at most a finite number of indices i for which fi(1) = 0 and with these factors omitted, the remaining product converges to a finite nonzero limit, and * f~ converges as N --+ 00. 11 - Write out (fl n i=l . -+ 00, n ) 2 ] -+ ) I. * . * f5)(2). a 03 the special role that . {fi(l)} plays? 12 Let { fi)gl be a sequence of arithmetic functions, none of which is identically zero. Also assume that f l * f 2 * - - - converges. Prove that f l * f 2 * # 0. Hint. Consider the f; for which fi(1) = 0.

Given n = P';'P;~ (each cllj 2 0), we take u 2 r and observe that n c (fl * . * fu)(n) = I-J j=1 n r U fj(pqj) . - . p:r fj(1)= j=r+l fj(Pj"'). j=l The infinite convolution product fi *f2 *. l - - - p:~, then ( f 1 * - - - * fu) ( n ) remains constant for all u 2 r. I f f = fi * f2 * , then with the preceding identity, r r j=1 j=1 f ( 4 = (fl * and f is multiplicative. Conversely, if f is multiplicative, then using the identity again, r r f ( n ) = U f ( P q ' ) = l-Ifj(P7j) = ( f l * * ' * * f v ) ( n ) j=l for all u 2 r , and thus f j=1 = fl t f2 * .