Incomplete Sums of Multiplicative Functions. I

A family of multiple integrals connected with relatives of the Dickman function

“…To deduce Lemma 1 from Lemma 2, we must show that the integrals arising from (10) are equivalent to those appearing in (2). Iterating the integral relation in (10), we find that…”

Jack van Lint (1932–2004): A survey of his scientific work

“…Starting in 1963 another series of papers on number theory was written, but these papers were definitely different from the previous ones. In [12,15,17,18,21,90] the asymptotic behavior of several number theoretic functions was discussed. Some of them, i.e.…”

“…Some of them, i.e. [12,15,17] were joint work with N.G. de Bruijn, while some of the others contained references to earlier results of de Bruijn.…”

“…They were defined by the sum over all n x of f (n, x) and f (n, n), respectively, where f (n, x) is the number of integers not exceeding x which are products of powers of prime factors of n. It was shown that F (x) ∼ G(x) as x → ∞. The next paper in this series was [15]. Let λ be a non-negative multiplicative function defined on the positive integers.…”

“…Rather strikingly, [129] constructed a quasi-residual 2-(28, 10, 45) design for which the Bruck-Ryser-Chowla theorem prevents the existence of any possible associated symmetric 2- (43,15,5) design. On the other hand, [70] earlier constructed two quasi-residual designs that are not residual, while [105] constructed infinite families of such designs.…”