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On some tauberian theorems related to the prime number theorem
Adolf Hildebrand, Gérald Tenenbaum
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Abstract
Here {cn}n=1 and {rn}n=0 are given sequences of complex numbers on which we shall impose some mild growth conditions. The equation (1.1) expresses each term of the sequence {an} as an average of the previous terms, weighted by the coefficients ck, plus a remainder term rn. One might expect that this repeated averaging process induces some degree of regularity on the behavior of an. We shall show that this is indeed the case, even if we impose no regularity conditions on the behavior of cn. Our motivation for this work came from two directions. The first is the theory of multiplicative arithmetic functions, where Wirsing [Wi], Halasz [Ha1, Ha2] and others have developed deep and powerful techniques to study the asymptotic behavior of the averages m(x) := e−x ∑ n ex f(n) of a multiplicative function f . As was observed by Wirsing, these averages satisfy integral equations of the type