Notes on the Riemann Hypothesis

These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture.Comment: 2 sections added, 55 pages, 6 figure

Relative FP-injective and FP-flat complexes and their model structures

In this paper, we introduce the notions of ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes in terms of complexes of type ${\rm FP}_n$. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes. We also introduce and study ${\rm FP}_n$-injective and ${\rm FP}_n$-flat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. The existence of pre-envelopes and covers in this setting is discussed, and we prove that any complex has an ${\rm FP}_n$-flat cover and an ${\rm FP}_n$-flat pre-envelope, and in the case $n \geq 2$ that any complex has an ${\rm FP}_n$-injective cover and an ${\rm FP}_n$-injective pre-envelope. Finally, we construct model structures on the category of complexes from the classes of modules with bounded ${\rm FP}_n$-injective and ${\rm FP}_n$-flat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences.Comment: 41 page