By P. T. Bateman, Harold G. Diamond

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**Example text**

29 (P. Erd6s). Let f be a real valued monotone multiplicative function. Iff is nondecreasing, then f = T"1 f o r some real nonnegative a. If f is nonincreasing then either f = T"1 for some real nonpositive cy or f = e + ce2 for some c E [0, I]. Proof. Suppose first that f is nondecreasing, so that f ( n ) 2 f(1) = 1 for all n. Let a be a fixed positive integer greater than 1. 13) For 21 = 1 these inequalities are trivial. It follows that for all integers w _> 1. Now let n be any integer exceeding a (still fixed), and define the integer r by a' 5 n < arS1.

Iff is nondecreasing, then f = T"1 f o r some real nonnegative a. If f is nonincreasing then either f = T"1 for some real nonpositive cy or f = e + ce2 for some c E [0, I]. Proof. Suppose first that f is nondecreasing, so that f ( n ) 2 f(1) = 1 for all n. Let a be a fixed positive integer greater than 1. 13) For 21 = 1 these inequalities are trivial. It follows that for all integers w _> 1. Now let n be any integer exceeding a (still fixed), and define the integer r by a' 5 n < arS1. Then T 2 1 and f ( a ' ) 5 f(n)5 f(art1), whence and therefore Letting n -+00, we get for each a 2 2.

C 1, 2. 3. ~ p ( n >P , := n ( z >:= set of primes, nsx 4. M ( 2 ) := C p ( n ) , 5. nsx nsx Except for the function f = 1, all of the preceding arithmetic functions are rather irregular. The r function, for example, satisfies ~ ( p " = ) a 1, from which we see that T is unbounded but assumes the value 2 infinitely often. On the other hand, a summatory function involves a large number of values of the associated arithmetic function. Thus we might hope that fluctuations of the arithmetic function are somehow smoothed out, enabling us to make statements about its average behavior.