The present note proves Riemann integrability of the $n$-th local modulus of continuity which is used within the definition of averaged moduli of smoothness ($\tau$-moduli). In addition it is shown that an averaged supremum norm ($\delta$-norm) can be calculated using the Riemann integral.

We prove a refined asymptotic formula for the number of representations of sufficiently large integer as a sum of a prime and a $k$-free number, $k \geqslant 2$.

We prove results analogous to certain theorems of Deshouillers and Iwaniec (Invent. Math. 70 (1982), 219-288]. Our proofs parallel theirs in the use made of the summation formulae of Bruggeman and Kuznetsov: where they require a lower bound on eigenvalues $\lambda_j = 1/4 + \kappa^2_j$ of the hyperbolic Laplacian operator (using that of Selberg) we need instead upper bounds on the moduli of the eigenvalues of a Hecke operator, obtaining these from recent work of Kim and Sarnak [J. Amer. Math. Soc. 16 (2003), 139-183]. Specifically, we give new bounds for sums $\sum_{Q/2\lt q\leqslant Q}\sum _{|\kappa_{j}|\leqslant K}|\sum_{N/2 \lt n \leqslant N}b_{n} \rho j(Dn)|^2$, where ($b_n$) is a complex sequence, and $j$ indexes the elements, $u_{j}(z)$, of a suitable orthonormal basis of the space spanned by the Maass cusp forms for the Hecke congruence subgroup $\Gamma_{0}(q)$, while $\rho_{j}(n)$ is the $n$-th Fourier coefficient at the cusp $\infty$ for $u_{j}(z)$, and $D$ is a large positive integer. Our bounds are strongest in cases where every prime factor of $D$ is a small power of $D$.

One application (briefly discussed in the paper) is a new mean-square bound for the modulus of a certain multiple sum involving Dirichlet characters modulo $D$. It is hoped this will be useful in the study of Carmichael numbers.