We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$. In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$.

In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$. This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of $\sum _{n\leqslant x}f(n)$.

The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ and the $q$th moment of a critical, approximately Gaussian, multiplicative chaos and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.

This article reviews a recent wave of literature on the resurgence of vinyl records, examining what it has claimed about vinyl's capacity for tangibility and the contrast to digital media, associated with intangibility. These claims are explained with reference to other literatures on touch, and it is suggested that vinyl's haptics mediates and embodies the emotionally rewarding production of a sense of self. The apparent contrast of vinyl aesthetics with classical music aesthetics is also discussed, and the presence of contemporary classical music within the vinyl resurgence is considered.

Halász’s theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\leqslant 1$ has either mean value $0$, or is ‘close to’ $n^{it}$ for some fixed $t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).

This article argues that the production and reception of certain recent electronic musics has resonated with criticisms of the perceived degenerative effects of digital technology on culture and ‘humanity’ – such as the lack of attention it promotes or the ‘information overload’ it causes – in an at least partially positive way. The resulting ambivalent aesthetics, sometimes thought of as one of ‘Internet music’, embraces particular negative notions of digital mediation in ways that can and have been thought of as satirical, exploratory or ‘accelerationist’. I examine three facets of this aesthetics: maximalism, kitsch and the uncanny valley. I also question the legitimacy of dramatising, even positively, digital media and culture as effectively ‘degenerate’.

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

‘THE PERCUSSION UNIVERSE OF AXEL BORUP-JØRGENSEN’: Solo; Music for percussion + viola; La Primavera; Periphrasis; Winter Music. Gert Mortensen (perc.), Percurama Percussion Ensemble, Tim Frederiksen (vla), Duo Crossfire, Michala Petri (rec.), DNSO Brass Quintet. OUR Recordings

The Kepler satellite provides a unique opportunity to study the detailed optical photometric variability of late-type stars with unprecedentedly long (several year) continuous monitoring and sensitivity to very small-scale variations. We are studying a sample of over two hundred cool (mid-A - late-K spectral type) stars using Kepler long-cadence (30 minute sampling) observations. These stars show a remarkable range of photometric variability, but in this paper we concentrate on rotational modulation due to starspots and flaring. Modulation at the 0.1% level is readily discernable. We highlight the rapid timescales of starspot evolution seen on solar-like stars with rotational periods between 2 and 7 days.

In this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

There has been significant anxiety among prescribers regarding the potential for cardiac adverse effects associated with acetylcholinesterase (AChE) inhibitors in Alzheimer's disease. There is no consensus on how to manage this cardiovascular risk, and memory clinics vary widely in their practice. Review of published evidence reveals that the incidence of cardiovascular side-effects is low, and that serious adverse events are rare. Intensive cardiovascular screening such as pre-treatment electrocardiograms or 24 h cardiac monitoring is not justified. Furthermore, there are no high-risk groups to target. This article suggests pragmatic guidelines for managing cardiovascular risk in patients receiving AChE inhibitors. The guidelines are intended to be easy to incorporate into routine clinical practice in a memory clinic.

During the past three years the measurement of stellar radial velocities has formed an important part of the spectroscopic programme of most observatories possessing large telescopes. As observations are carried to fainter and fainter stars and the number of observable objects increases rapidly, a natural development has been the selection of special groups and types of stars, the radial velocities of which will aid in the solution of certain specific problems. Illustrations are the studies of the O, B and A type stars made at the Dominion Astrophysical, the Lick, and the Simeis Observatories, of the members of the galactic clusters at the Lick Observatory, and of the fainter Cepheid variables and early-type stars with strong interstellar lines at the Mount Wilson Observatory.

Since the Paris meeting of the Union 3050 parallax plates have been secured and in the same interval 5604 plates have been measured and 182 parallaxes have been determined.

In November 1934 the President circulated a letter to the members of the Commission as follows:

Since the 1932 meeting the following projects have been completed, or are nearing completion:

1. (1) The publication of many lists of trigonometric parallaxes.

2. (2) The determination of the spectroscopic parallaxes of 4179 stars at Mt Wilson Observatory by Adams, Joy and Humason.

3. (3) A discussion of systematic errors of trigonometric parallaxes by van Maanen and a re-discussion in the Astrophysical Journal of the same material by Mitchell and by Sterne.

4. (4) The compilation of a second Yale Catalogue to include parallaxes completed before the end of 1934.

5. (5) Substantial progress on the proper motions of 32,000 stars by Boss and his associates at the Dudley Observatory.

6. (6) The publication at the Radcliffe Observatory of the proper motions of 32,000 stars from photographs on 115 Selected Areas.

7. (7) The completion of the dynamical parallaxes of 2000 stars.

8. (8) The completion of the proper motions of 18,000 stars derived from parallax plates at the Leander McCormick Observatory.

9. (9) The publication at the Yale Observatory of the proper motions of 40,000 stars with a probable error less than 0”.010 determined from photographs by re-observing in zones the Astronomische Gesellschaft stars.

10. (10) The determination of the proper motions of 50,000 stars in the Southern Hemisphere by Luyten from Harvard photographs.

The three years that have elapsed since the Harvard meeting of the Union have witnessed steady progress in the determination of radial velocities. While the three large Pacific Coast Observatories have naturally been able to make the greatest additions to radial velocity work, the Yerkes Observatory, the Simeiz Observatory and the Observatory of the University of Michigan have also made valuable contributions. It is a pleasure to report that there will soon be three major accessions to the list of observatories capable of determining radial velocities. The David Dunlap Observatory of the University of Toronto with its 74-inch telescope, which should be in operation soon after the meeting, will have radial velocities as a prominent feature of its programme. The McDonald Observatory of the University of Texas with an 80-inch telescope now under construction should be ready to commence operations in 1936 and will undertake an extensive radial velocity programme. The Radcliffe Observatory at Oxford has now been granted permission by the Courts to remove to Pretoria, South Africa, and will establish there a 74-inch reflecting telescope, which will also be largely employed in the determination of the urgently needed radial velocities of the southern stars fainter than 5.5 visual magnitude. The Commission may, I believe, congratulate itself that substantial assistance in the preliminary steps leading to this permission of removal was provided by our action at the last meeting in presenting a resolution to the Union, duly passed by the General Assembly, pointing out the urgent need for additional radial velocities in the southern sky, and strongly supporting the project of the Radcliffe Observatory to establish a large telescope at Pretoria.