Sums of Steinhaus random multiplicative functions over short intervals [x, x+y] (y→∞, y=o(x)) have Gaussian limiting distributions after a normalization that is not √y when y is close to x.
On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
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abstract
We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as $T \leq t \leq 2T$ varies. In particular, we prove that for $(1+o(1))T$ values of $T \leq t \leq 2T$ we have $\max_{|h| \leq 1/2} \log|\zeta(1/2+it+ih)| \leq \log\log T - (3/4 + o(1))\log\log\log T$, matching for the first time with both the leading and second order terms predicted by a conjecture of Fyodorov, Hiary and Keating. The proofs work by approximating the zeta function in mean square by the product of a Dirichlet polynomial over smooth numbers and one over rough numbers. They then apply ideas and results from corresponding random model problems to compute averages of this product, under size restrictions on the smooth part that hold for most $T \leq t \leq 2T$ (but reduce the size of the averages). There are connections with the study of critical multiplicative chaos. Unlike in some previous work, our arguments never shift away from the critical line by more than a tiny amount $1/\log T$, and they don't require explicit calculations of Fourier transforms of Dirichlet polynomials.
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2026 5verdicts
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Proves that sum of Steinhaus random multiplicative function over A converges to CN(0,1) only if |A|=o(N), with sharpness for most sets of density ρ where (1-ρ)^{-1}=o((log log N)^{1/2}).
Formulates an avatar of the Fyodorov-Hiary-Keating conjecture for black hole microstate counts, implying sharp bounds on CFT primary operator interval counts and suggesting that AdS spectra exhibit extreme value statistics of Gaussian log-correlated random matrices.
Establishes sharp lower bounds matching prior upper bounds for moments of short character sums, zeta sums, and twisted sums with multiplicative weights, for x up to r^0.499.
Under RH, the measure of t in [T,2T] with |zeta(1/2+it)| > (log T)^k is <= C_k (log T)^{-k^2}/sqrt(log log T) with C_k=exp(e^{ck}), implying 2k-moment bounds C_k (log T)^{k^2}.
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