Under the Riemann Hypothesis, we show that as t varies in T⩽t⩽2T, the distribution of log|ζ(1/2+it)| with respect to the measure |ζfalse(1/2+itfalse)|2dt is approximately normal with mean loglogT and variance 12loglogT.

We prove a central limit theorem for log ⁡ | ζ ⁢ ( 1 2 + i ⁢ t ) | {\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure | ζ ( m ) ⁢ ( 1 2 + i ⁢ t ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} ( k , m ∈ ℕ {k,m\in\mathbb{N}} ), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure | ζ ⁢ ( 1 2 + i ⁢ t + i ⁢ α ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta(\frac{1}{2}+it+i\alpha)\rvert^{2k}\,dt} , with α ∈ ( - 1 , 1 ) {\alpha\in(-1,1)} . Finally, we prove unconditionally the analogue result in the random matrix theory context.

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields ${\mathbb {Q}}(\sqrt {d+1}),\dots ,{\mathbb {Q}}(\sqrt {d+k})$ have class numbers essentially as large as possible.

We prove a central limit theorem for log |ζ(1/2 + it)| with respect to the measure |ζ (m) (1/2 + it)| 2k dt (k, m ∈ N), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure |ζ(1/2 + it + iα)| 2k dt, with α ∈ (−1, 1). Finally we prove unconditionally the analogue result in the random matrix theory context.

We study the moments of L-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a long Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.