Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

Let |θ| < π/2 and . By refining Selberg's method, we study the large values of as t → ∞ For σ close to ½ we obtain Ω+ estimates that are as good as those obtained previously on the Riemann Hypothesis. In particular, we show that

and

Our results supplement those of Montgomery which are good when σ > ½ is fixed.

The best current bounds for the proportion of zeros of ζ(s) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the critical line. To apply Levinson's method one first needs an asymptotic formula for the meansquare from 0 to T of ζ(s)M(s) near the -line, where

where μ(n) is the Möbius function, h(x) is a real polynomial with h(0) = 0, and y=Tθ for some θ > 0. It turns out that the parameter θ is critical to the method: having an asymptotic formula valid for large values of θ is necessary in order to obtain good results. For example, if we let κ denote the proportion of nontrivial zeros of ζ(s) which are simple and on the critical line, then having the formula valid for 0 < θ < yields κ > 0·3562, having 0 < θ < gives κ > 0·40219, and it is necessary to have θ > 0·165 in order to obtain a positive lower bound for κ. At present, it is known that the asymptotic formula remains valid for 0 < θ < , this is due to Conrey. Without assuming the Riemann Hypothesis, Levinson's method provides the only known way of obtaining a positive lower bound for κ.

A well-known theorem of Hardy and Littlewood gives a three-term asymptotic formula, counting the lattice points inside an expanding, right triangle. In this paper a generalisation of their theorem is presented. Also an analytic method is developed which enables one to interpret the coefficients in the formula. These methods are combined to give a generalisation of a “heightcounting” formula of Györy and Pethö which itself was a generalisation of a theorem of Lang.