Ayesha Irfan

Almost primes in short intervals and long Dirichlet polynomials

The k-fold von Mangoldt convolution $\Lambda^{*k}$ ties together the 2k-th moments of the logarithmic derivative of the Riemann zeta function and the short-interval variance of weighted almost-prime counts. We study these moments through mean squares of long Dirichlet polynomials. Assuming the Riemann Hypothesis and a suitable shifted-moment conjecture, we show that their leading coefficient differentiates to give the leading coefficient of the almost-prime variance. I will outline both arguments and explain how the ratios conjecture motivates the required moment prediction. Following the precedent of the k-th divisor problem, we formulate a common unitary matrix model for these quantities through function-field results. Their asymptotic analysis reduces to geometric problems whose dimensions depend only on k. Approximating these counts by integrals proves the existence of the limiting matrix coefficient and gives explicit piecewise-polynomial formulae. From this, we obtain conjectures for the asymptotic for almost primes in short intervals.