This paper proves new, explicit lower bounds for several high moments of the Riemann zeta function. The authors develop asymptotic formulae for two kinds of averaged values that attach a short “amplifier” to the zeta function. From those amplified averages they deduce unconditional numerical lower bounds for ordinary and joint moments that closely match predictions from random matrix theory and earlier conjectures of Keating, Snaith and Wei.

What the researchers did. They study integrals of powers of |ζ(1/2+it)| and its derivatives over long t-intervals. To probe large values they insert a short Dirichlet polynomial A(s) — the amplifier — and work with averages of the form ∫|ζ(1/2+it)|^2 |A(1/2+it)+χ(1/2+it)A(1/2−it)|^{2k} dt and the analogous fourth-moment twisted average. Here χ is the usual factor that appears in the zeta functional equation. The new ingredient is a “two-piece” amplifier (the A + χA(1−s) combination) and a careful choice of polynomial coefficients for A(s). The authors obtain asymptotic formulae for these amplified second and fourth moments when the amplifier is short enough; the allowed amplifier length shrinks as the amplifier power k grows.

Why it matters. From the amplified asymptotics the paper extracts effective, unconditional lower bounds for many joint integer moments. Notably they prove for the sixth moment that M3(T) = ∫_0^T |ζ(1/2+it)|^6 dt ≥ (34.1+o(1)) c3 T (log T)^9. This improves earlier unconditional bounds and is close to the Keating–Snaith conjectured value 42 c3 T (log T)^9. The authors also give improved lower bounds for moments of ζ′ and ζ′′ and for many mixed moments such as ∫|ζ|^2|ζ′|^4 and ∫|ζ|^4|ζ′|^2, always with explicit numerical constants to compare with conjectures.

Other results and methods. By modifying prior work of Soundararajan the paper removes the conditional assumption known as the Lindelöf Hypothesis in several earlier results. It also extends and sharpens Soundararajan’s lower bounds for moments k = 5,6 and supplies stronger unconditional bounds up to k = 11 by adding a second piece to the amplifier. The authors give a general theorem that produces lower bounds for any joint integer moment; the main arithmetic factor C(a1,...,aK) that appears is shown to be positive and is expressed in terms of combinatorial derivatives and an Euler product.