Some new aspects and analytic technical applications of diophantine approximations (Q752754)

This expository paper, dedicated to the memory of L. Collatz, deals with certain sharp asymptotical problems arising from diophantine approximations. The topics are connected also with analytic theory of numbers, higher zeta-functions, generalized integro-differential operators, as well as with some applications in telecommunication technology. The starting-point is a famous question due to Koksma (1947), answered partly by Erdős and completely by Gál in 1949, which was later extended and investigated further by the author, Carlitz, Mordell and others in several directions, until recent years. It can be formulated as follows: find such a close upper bound for the ``error square integral'' \[ \int^1_0[\sum^N_{\nu =1}(<n_{\nu}u>- 1/2)]^2\,du,\quad <x>=x-[x] \] \((n_1<n_2< \ldots <n_N\) denoting an arbitrary system of positive integers) which depends only on \(N\) and is independent of the special choice of the \(n_{\nu}\). This so-called ``estimate problem of Hardy and Littlewood'' admits strong generalizations in as much as the function \(x-1/2\) is replaced by Bernoulli polynomials of higher degree, or (more generally) by appropriate values of the Hurwitz and Lerch zeta-functions, the latter being defined by analytic continuation of the series \[ \sum^{\infty}_{m=0}(x+m)^{- s},\quad \sum^{\infty}_{m=0}(x+m)^{-s}e^{2m\pi i\omega},\quad \operatorname{Re} s>1, \] with respect to the complex variable \(s\). Since the Hurwitz zeta-function \(\zeta(s,x)\) occurs as ``kernel function'' also in H. Weyl's definition of ``fractional integral'', we get still new relations belonging to the theory of integro-differential operators of nonintegral order. The technical applications bear mainly on importance of the (1-periodic) function \(1/2-<x>\) and of its pendants in electrical engineering. (Cf. e.g. some works of Van der Pol since the fifties.) Note that the bibliography at the end of the paper contains nearly 40 items.