We investigate the distribution of the Riemann zeta-function on the line Re(s) = σ. For 1 2 < σ ≤ 1 we obtain an upper bound on the discrepancy between the distribution of ζ(s) and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of ζ(s) inside of the critical strip, strengthening a previous result of the first author.As an application of these results we obtain the first effective error term for the number of solutions to ζ(s) = a in a strip 1 2 < σ 1 < σ 2 < 1. Previously in the strip 1 2 < σ < 1 only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg) and to the right of the abscissa of absolute convergence (due to Matsumoto). In general our results are an improvement of the classical Bohr-Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function.

Abstract. We study the small scale distribution of the L 2 mass of eigenfunctions of the Laplacian on the flat torus T d . Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose L 2 mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions d = 3, 4 we can restrict to individual eigenspaces and show small scale equidistribution in that context.We also study irregularities of quantum equidistribution: We construct eigenfunctions whose L 2 mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension d = 4 we show the existence of eigenfunctions for which the proportion of L 2 mass in small balls blows up at certain scales.

Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c(n) denote the nth Fourier coefficient of g, normalized so that c(n) is real for all $$n \ge 1$$ n ≥ 1 . A theorem of Waldspurger determines the magnitude of c(n) at fundamental discriminants n by establishing that the square of c(n) is proportional to the central value of a certain L-function. The signs of the sequence c(n) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that $$c(n) < 0$$ c ( n ) < 0 and respectively $$c(n) > 0$$ c ( n ) > 0 holds for a positive proportion of fundamental discriminants n. Moreover we show that the sequence $$\{c(n)\}$$ { c ( n ) } where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c(n) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4N with N odd, square-free.

We study the behavior of zeros and mass of holomorphic Hecke cusp forms on SL 2 (Z)\H at small scales. In particular, we examine the distribution of the zeros within hyperbolic balls whose radii shrink sufficiently slowly as k → ∞. We show that the zeros equidistribute within such balls as k → ∞ as long as the radii shrink at a rate at most a small power of 1/ log k. This relies on a new, effective, proof of Rudnick's theorem on equidistribution of the zeros and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper.We also examine the distribution of the zeros near the cusp of SL 2 (Z)\H. Ghosh and Sarnak conjectured that almost all the zeros here lie on two vertical geodesics. We show that for almost all forms a positive proportion of zeros high in the cusp do lie on these geodesics. For all forms, we assume the Generalized Lindelöf Hypothesis and establish a lower bound on the number of zeros that lie on these geodesics, which is significantly stronger than the previous unconditional results.