  We show that a very simple randomised algorithm for numerical integration can
produce a near optimal rate of convergence for integrals of functions in the
$d$-dimensional weighted Korobov space. This algorithm uses a lattice rule with
a fixed generating vector and the only random element is the choice of the
number of function evaluations. For a given computational budget $n$ of a
maximum allowed number of function evaluations, we uniformly pick a prime $p$
in the range $n/2 < p \le n$. We show error bounds for the randomised error,
which is defined as the worst case expected error, of the form $O(n^{-\alpha -
1/2 + \delta})$, with $\delta > 0$, for a Korobov space with smoothness $\alpha
> 1/2$ and general weights. The implied constant in the bound is
dimension-independent given the usual conditions on the weights. We present an
algorithm that can construct suitable generating vectors \emph{offline} ahead
of time at cost $O(d n^4 / \ln n)$ when the weight parameters defining the
Korobov spaces are so-called product weights. For this case, numerical
experiments confirm our theory that the new randomised algorithm achieves the
near optimal rate of the randomised error.
