Class number one criterion for some non-normal totally real cubic fields

Let ${\{K_m\}_{m\geq 4}}$ be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial $f_m(x)=x^3-mx^2-(m+1)x-1$, where $m$ is an integer with $m\geq 4$. In this paper, we will give a class number one criterion for $K_m$.Comment: 9 page

Feynman's sunshine numbers

This is an expansion of a talk for mathematics and physics students of the Manchester Grammar and Manchester High Schools. It deals with numbers such as the Riemann zeta value zeta(3)=sum_{n>0}1/n^3. Zeta values appear in the description of sunshine and of relics from the Big Bang. They also result from Feynman diagrams, which occur in the quantum field theory of fundamental particles such as photons, electrons and positrons. My talk included 7 reasonably simple problems, for which I here add solutions, with further details of their context.Comment: 24 pages, LaTe

On the Holography of Free Yang-Mills

We study the AdS$_5$/CFT$_4$ duality where the boundary CFT is free Yang-Mills theory with gauge group SU(N). At the planar level we use the spectrum and correlation functions of the boundary theory to explicate features of the bulk theory. Further, by computing the one-loop partition function of the bulk theory using the methods of arXiv:1603.05387, we argue that the bulk coupling constant should be shifted to $N^2$ from $N^2-1$. Similar conclusions are reached by studying the dualities in thermal AdS$_5$ with $S^1\times S^3$ boundary.Comment: 44 pages, version to appear in JHE

Nonvanishing of twists of LL-functions attached to Hilbert modular forms

We describe algorithms for computing central values of twists of $L$-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and predictions from random matrix theory.Comment: 19 page