Correlations of the Moebius and Liouville functions and the twin prime conjecture

In this note we describe weight functions that exhibit a transitional behavior between weak and strong correlation with the Liouville function. We also describe a binary problem which may be considered as an interpolation between Chowla's conjecture for two-point correlations of the M\"obius function and the twin prime conjecture, in view of recent parity breaking results of K. Matom\"aki, M. Radziwi{\l}{\l} and T. Tao

Terevaka Archaeological Outreach (TAO) 2022–2023 Field Report: Program Expansion

Since 2003, Terevaka Archaeological Outreach (TAO) has offered unforgettable educational opportunities regarding archaeology, traditional lifeways, technology, and sustainable development at no cost for Rapa Nui high school students. After 20 years of refining our pedagogy and implementation, 2022–2023 marked a year of rapid expansion for TAO. Within a 10-month period, TAO completed two projects on Rapa Nui, and also launched sister programs in the Sacred Valley of Peru and in Patagonia National Park of Chile

Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao

We determine for what proportion of integers $h$ one now knows that there are infinitely many prime pairs $p,\ p+h$ as a consequence of the Zhang-Maynard-Tao theorem. We consider the natural generalization of this to $k$-tuples of integers, and we determine the limit of what can be deduced assuming only the Zhang-Maynard-Tao theorem.Comment: 9 pages. Final version. Some minor changes, Analytic Number Theory - In Honor of Helmut Maier's 60th Birthday, Springer, 201

Fuglede's conjecture fails in dimension 4

In this note we give an example of a set \W\subset \R^4 such that L^2(\W) admits an orthonormal basis of exponentials \{\frac{1}{|\W |^{1/2}}e^{2\pi i x, \xi}\}_{\xi\in\L} for some set \L\subset\R^4, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.Comment: 6 page