An improvement of an inequality of koksma

AbstractAn expression for the Koksma integral [Indag. Math.13 (1951), 285-287;MR13, 539] is given. The result is applied to show that the constant 1/3 in Koksma's inequality [ibidem] can be replaced by 1/4 and this constant is the best possible. Also the mean value of the integral is determined

Distribution functions of the sequence phi(n)/n, n in (k,k+N]

International audienceIt is well known that the sequence $\varphi(n)/n$, n=1,2,... has a singular asymptotic distribution function. P. Erdös in 1946 found a sufficient condition on the sequence of intervals (k,k+N], such that phi(n)/n, n in (k,k+N], has the same singular function. In this note we prove a sufficient and necessary condition. For simplifying the necessary condition we express the sum \sum_{k n*k+N(!(n) ¡ log logN)2, where !(n) is the number of di®erent primes divided n

The asymptotic distribution function of the 4-dimensional shifted van der Corput sequence

Abstract Let γq(n) be the van der Corput sequence in the base q and g(x, y, z, u) be an asymptotic distribution function of the 4-dimensional sequence In this paper we find an explicit formula for g(x, x, x, x) and then as an example we find the limit for the base q = 4, 5, 6, . . . Also we find an explicit form of sth iteration T(s)(x) of the von Neumann-Kakutani transformation defined by T(γq(n)) = γq(n + 1)