Optimal oscillations and the mean of the fourth moment of L(1,χ), with applications to cotangent sums, Hardy sums and a random sieve

Let E₄(p) be the error term in the fourth moment of L(1,ψ) over the odd characters ψ modulo a prime p; by Walum's identity it is, up to the factor p/π⁴, the error term in the mean square of the Dedekind sums s(h,p). We prove that E₄(p) = Ω±(exp(c log p / log log p)), so that Zhang's upper bound is optimal up to the constant in the exponent, and that the mean of |E₄(p)| over the primes p ≤ X is O((log X)⁴). Reading E₄(p) as a Riemann sum along the shifted grids {(t+ϑ)/p}, we determine on average over p each of its Fourier modes, through the uniform Titchmarsh divisor theorem of Assing, Blomer and Li. This gives the mean of E₄(p) smoothed at the relative scale (log p)^(−C), and we conjecture that without smoothing the mean is −(π²/4) log²p + (π²/2)(ζ′(2)/ζ(2) − γ) log p + 1.7333…, in agreement with the exact values. Through an exact identity, the error term E(q) in the second moment of the cotangent sum c₀(a/q) of Bettin and Conrey is Ω±(q exp(c log q / log log q)), and conjecturally E(q) ≈ −(1/π²) q log²q on average; numerically E(q) < 0 at every prime 101 ≤ q ≤ 2·10⁵. This contradicts a theorem of Maier and Rassias, according to which E(q) ≥ C q log²q for all large primes q. The same methods give the maximal order, the average size and the smoothed means of a mean value of the Hardy–Berndt sums 2s(2h,p) − s(h,p) weighted by Selberg's sieve function, and we record their consequences for the parity-marked pair covariances of a random sieve. This paper collects the new results of doi:10.5281/zenodo.23613561, with complete proofs of the results on the fourth moment and on the cotangent sums.