On the moments of the moments of ζ(1/2 + it)

Maxima of log-correlated fields: some recent developments

“…, h k ) with h j ∈ R. Then One may note that the structure of (280) is very similar to the multiple contour integral appearing in lemma 3.12, used to prove theorem 3.6. The main result of [17] is the following. where the term α k,β = A kβ (0, .…”

“…In [77], Fyodorov and Bouchard showed that in the large M limit, the (positive, integer) moments 17 of the partition function…”

“…We now review the relevant number theoretic concepts, beginning with the definition of one of the central functions of number theory. In general, expressions of the type (17) ∞ n=1 a n n s for s, a n ∈ C are known as Dirichlet series. Hence, ζ(s) can be defined by a Dirichlet series with a n = (1, 1, 1, .…”

“…Most broadly, an L-function is a Dirichlet series with an Euler product and a functional equation (cf. (17), (18), and (19)). In order to keep this review self-contained we focus on two canonical extensions.…”

“…where ½ p∤∆ is 1 for the 'good primes' not dividing the discriminant, and 0 otherwise. We shall focus in this section on the unitary case, noting that there are more general connections between more general number theoretic functions -L-functions -and the other classical compact matrix groups (see [17,52,103,104,109,110,118,132]), the discussion of which we postpone until later. For brevity, we temporarily write P N ≡ P U(N ) for the characteristic polynomial of a matrix in the CUE.…”