Almost primes in short intervals and averages of long Dirichlet polynomial approximations to $(ζ'/ζ)^{k}$

Abstract

We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for $\bigl(ζ'/ζ\bigr)^{k}$ on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function $Λ^{*k}$ in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of $Λ^{*k}$ over monic polynomials in $\mathbb F_q[T]$; we denote its leading coefficient by $δ_k(c)$, where the total trace degree and matrix size grow with a fixed ratio $c$. Under a weaker shifted moments conjecture and RH, we recover this $δ_k(c)$ as the leading coefficient of the variance of short-interval sums of $Λ^{*k}$ over the integers. It is the counterpart of $γ_k(c)$, the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of $ζ(s)^k$ replaced by those of $\bigl(-ζ'/ζ\bigr)^k$.