We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the abc-conjecture.

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape (Formula presented.) for a suitable constant (Formula presented.) and a polynomial (Formula presented.). Such examinations are performed both unconditionally and under the assumption of a weaker version of the (Formula presented.) -conjecture.

We compute a collection of mixed moments of the Riemann-zeta function by means of an approximate functional equation for the product of zeta functions. As an application we provide estimates for three-dimensional exponential sums with monomials which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.