Abstract We investigate when the exponential sum is bounded, for a multiplicative function and . We show that under natural assumptions, is bounded only when is very close to a twisted Dirichlet character . We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when is completely multiplicative and is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of .
Abstract We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations compared to previous results. This is the first approach to the murmuration phenomenon using a relative trace formula, showing its robustness.
Abstract This paper answers a long‐standing open problem concerning the completeness of the metric space of asymmetry classes introduced by Ewald and Shephard (1966). We prove that this metric space is incomplete for . Our approach reveals that the Ewald–Shephard metric is equivalent to a quotient metric on a certain function space of odd continuous functions modulo linear functions, and that the Lipschitz regularity of odd parts of support functions constitutes an essential obstruction to completeness. We explicitly construct a Cauchy sequence of asymmetry classes whose limit cannot be represented by any compact convex body, thereby rigorously establishing the incompleteness of the space.
Abstract In number theory, many major results related to the additive properties of primes are proven using the methods of sieve theory. However, in nearly every case, the existing proofs of these results are ineffective, in that explicit values for which they hold cannot be computed. The reason for this ineffectivity is the reliance on the Bombieri–Vinogradov theorem. In this paper, we show that any classical sifting problem with a Bombieri–Vinogradov style error term can, in fact, be made effective, with no loss to the asymptotic form of the original (ineffective) result. This is done by carefully modifying the sieve upper and lower bounds to avoid the usual complications regarding Siegel zeros. We also provide some simple applications. For example, we show that one may effectively bound the number of primes such that is also prime by where
In this paper, we study the values of modular functions at real quadratic irrationalities which are defined in terms of cycle integrals of modular functions associated with closed geodesics. We refine a result of Bengoechea and Imamogllu to include values (and its limiting behaviour) of modular functions at a larger class of real quadratic irrationalities. As a consequence, we deduce the limiting behaviour of values of modular functions at powers of Pisot-Vijayaraghavan numbers. Further, we deduce distribution of class numbers of real quadratic fields having discriminants of a particular type.
