A conjectural asymptotic formula for multiplicative chaos in number theory

Almost sure lower bounds for a model problem for multiplicative chaos in number theory

“…Since there are at most 2𝑝 blocks, we therefore obtain, for any 𝑖, that 𝑖 + 𝑒 −ℎ 𝑖+1 𝟙(ℎ 𝑖+1 ≠ log 𝐾) + ⋯ + 𝑒 −ℎ 𝑗−1 𝟙(ℎ 𝑗−1 ≠ log 𝐾) Summing this in turn over 𝑖 gives a bound ⩽ 4𝑝 log(2𝑝) + 𝑂(𝑝). Including the range where 𝜋 < 𝑡 𝜎(𝑗) − 𝑡 𝜎(𝑖) < 2𝜋 and the factor1 𝜋 , this shows that the expression in (5.7) isSo far, we have therefore shown that the expectation of the expression in (5.6) is…”

“…Finally, we also want to point the reader towards the work [1], where the authors numerically study the asymptotics of the first moment of $$A(n)$$ (which are not known to exist) by developing an algorithm that allows a more effective computation.…”

“…Let (𝑋(𝑘)) 𝑘⩾1 be a sequence of independent standard complex Gaussian random variables, so that their real and imaginary parts are independent, centered and have variance 1 2 . Consider the sequence of random variables (𝐴(𝑛)) 𝑛⩾0 defined through the following formal identity of power series:…”