It is proved that as , uniformly for all positive integers , we have where . Here, is the Dickman function. We have  and  when , which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet -functions. On the other hand, when assuming the Riemann hypothesis and the generalized Riemann hypothesis, we establish upper bounds for and . Furthermore, when assuming the Granville-Soundararajan conjecture is true, we establish the following asymptotic formulas: where is prime and is given.

Haar null sets were introduced by Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, Darji defined a categorical version of Haar null sets, namely Haar meagre sets. The present paper aims to show that, whenever is a closed, convex subset of a separable Banach space, is Haar null if and only if is Haar meagre. We then use this fact to improve a theorem of Matoušková and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation of separable Banach lattices whose positive cone is not Haar null.

We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free.

We employ some results about continued fraction expansions of Herglotz-Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral problem through explicit formulas.

We study global injectivity of proper branched coverings from the open Euclidean -ball onto an open subset of the Euclidean -space in the case where the branch set is compact. In particular, we show that such mappings are homeomorphisms when or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen.

We prove new upper bounds on the number of representations of rational numbers as a sum of four unit fractions, giving five different regions, depending on the size of in terms of . In particular, we improve the most relevant cases, when is small, and when is close to . The improvements stem from not only studying complete parametrizations of the set of solutions, but simplifying this set appropriately. Certain subsets of all parameters define the set of all solutions, up to applications of divisor functions, which has little impact on the upper bound of the number of solutions. These 'approximate parametrizations' were the key point to enable computer programmes to filter through a large number of equations and inequalities. Furthermore, this result leads to new upper bounds for the number of representations of rational numbers as sums of more than four unit fractions.

About a decade ago, it was realised that the satisfaction of a given (or ) of the form in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called , and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a in any arbitrary non-trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form , which we call . We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non-trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo-loop conditions of finite width, a generalisation suitable for non-idempotent algebras; they are of the form , and of central importance for the structure of algebras associated with -categorical structures. We show that for the latter, satisfaction of a pseudo-loop condition is characterised by , that is, loops modulo the action of the automorphism group, and that a weakest pseudo-loop condition exists (for -categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non-trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.

We prove several unique continuation results for biharmonic maps between Riemannian manifolds.