Hardy sums weighted by Selberg's function, the fourth moment of L(1,χ), and the parity floor of a random sieve

We show that the error term in the asymptotic formula for the mean square of the Dedekind sums s(h,p) modulo a prime p is Ω±(p·exp(c·log p/log log p)) for some c > 0, so that Zhang's upper bound for it is optimal up to the constant in the exponent. Equivalently, the error term in the fourth moment of L(1,χ) over the odd characters modulo p has maximal order exp(Θ(log p/log log p)), with both signs, although the mean of its absolute value over the primes is at most polylogarithmic. The same holds for a twisted fourth moment, a mean value of Hardy–Berndt sums weighted by the function h(d) of Selberg's sieve. Both quantities are Riemann sums of products of Davenport series. Using the uniform Titchmarsh divisor theorem of Assing, Blomer and Li, we compute the mean over the primes of each of their Fourier modes, up to a small power of p, and hence their mean values after a slight smoothing. Without the smoothing the mean values are stated as conjectures, equivalent to a regularity property; for the fourth moment the conjectured mean agrees with exact computations at 3691 primes up to about 10^7, with no free parameter. In the second part we apply these results to a random sieve, the Cramér–Hardy–Littlewood model, and to the sum of its pair covariances with a parity mark. Under explicit hypotheses on the means and the variances of its one-prime blocks and on its blocks with several primes, this sum converges in mean to e^(−2γ)/8. The hypothesis on the variances is proved in a periodic model and, for the moduli up to 10p, in a range of primes, and the part of the hypothesis on the means that concerns the floor follows from a conjectured law for the weighted Hardy–Berndt sums.