Extremal values of Δ(x,N)=Σn<xN(n,N=1)1-xφ(N)

The function Delta(x, N) as defined in the title is closely associated via Delta(N) = sup(x) \Delta(x, N)\ to several problems in the upper bound sieve. It is also known via a classical theorem of Franel that certain conjectured bounds involving averages of Delta(x, N) are equivalent to the Riemann Hypothesis. We improve the unconditional bounds which have been hitherto obtained for Delta(N) and show that these are close to being optimal. Several auxiliary results relating Delta(Np) to Delta(N), where p is a prime with p inverted iota N, are also obtained and two new conjectures stated.