Primes and prime ideals in short intervals

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.Comment: minor change to Proposition

Global correlates of terrestrial and marine coverage by protected areas on islands

Many islands are biodiversity hotspots but also extinction epicenters. In addition to strong cultural connections to nature, islanders derive a significant part of their economy and broader wellbeing from this biodiversity. Islands are thus considered as the socio-ecosystems most vulnerable to species and habitat loss. Yet, the extent and key correlates of protected area coverage on islands is still unknown. Here we assess the relative influence of climate, geography, habitat diversity, culture, resource capacity, and human footprint on terrestrial and marine protected area coverage across 2323 inhabited islands globally. We show that, on average, 22% of terrestrial and 13% of marine island areas are under protection status, but that half of all islands have no protected areas. Climate, diversity of languages, human population density and development are strongly associated with differences observed in protected area coverage among islands. Our study suggests that economic development and population growth may critically limit the amount of protection on islands. © 2020, The Author(s)