### A Brun-Titschmarsh theorem for multiplicative functions.

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The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.

The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.

The main purpose of this paper is to study a hybrid mean value problem related to the Dedekind sums by using estimates of character sums and analytic methods.

This article is concerned with estimations from below for the remainder term in Weyl’s law for the spectral counting function of certain rational (2ℓ + 1)-dimensional Heisenberg manifolds. Concentrating on the case of odd ℓ, it continues the work done in part I [21] which dealt with even ℓ.