Moments of the Riemann zeta function on short intervals of the critical line

Low pseudomoments of the Riemann zeta function and its powers

“…On the range T ≫ 1 on the other hand there are only large primes in the sense that no Euler product factors are expected to be constant over the whole range of integration. These ideas strongly resemble observations made in [AOR19]. There, the authors analyse the behaviour of…”

On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture

“…And for fixed β < 1 one expects to find many t with zeta this large in each bounded interval, whilst for fixed β > 1 one expects to find no such t in most intervals of bounded length. This explains the so-called freezing transition in the integrals 1/2 −1/2 |ζ(1/2 + it + ih)| 2β dh, conjectured by Fyodorov-Keating [8] and Fyodorov-Hiary-Keating [7], and proved up to factors log ǫ T by Arguin, Ouimet and Radziwi l l [4]. The transitional case where β = 1 is thus rather delicate, and the one which really captures information about the typical behaviour of the local maximum max |h|≤1/2 |ζ(1/2 + it + ih)|.…”

The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan]

“…where the sum is over prime powers p k . Here W = (log log log T ) 4 , and P = T 1/(log log log T ) 2 , and the o(1) term tends to 0 as T → ∞.…”

“…We also note that some parts of Fyodorov, Hiary and Keating's conjectures in the random matrix setting, and about the partition function t+1 t−1 |ζ(1/2 + iw)| 2β dw, have recently been proved using ideas related to those we shall describe here. See the papers [1,4,17], for example. Conjecture 3.2 suggests that our earlier heuristic analysis isn't quite right, but almost, since the first order term log log T that we obtained was the same.…”