Search Results

Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define $$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$ for all real numbers $x \in …

I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ple …

I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures …

Let $X = Hom(T, G_m)$ and $Y = Hom(G_m, T)$ be the character and cocharacter lattices of $T$, respectively. Let $k$ be the (algebraically closed) ground field. Note that $T = Spec(k[X])$ (a canonica …

The Langlands correspondence for higher local fields is still at an early stage of development. I haven't really kept up with it, but here's some key points. As the question stated, and Loren commen …

In the case $p \equiv 1$ mod $4$, the connection is to the real quadratic field ${\mathbb Q}(\sqrt{p})$, whereas the case $p \equiv 3$ mod $4$ is connected to the imaginary quadratic field ${\mathbb Q …

I was wondering about the distribution of $\sqrt{p}$ mod $1$ this morning, as one does while brushing one's teeth. I remembered the paper of Elkies and McMullen (Duke Math. J. 123 (2004), no. 1, 95–1 …

Here are some more details. As John Voight said, the quaternion algebra is kind of irrelevant here. If $\gamma$ is a regular semisimple element, then its centralizer is a torus ${\mathbf T}$ over ${ …

For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.) I am c …

There's no more serious name for the topograph, as far as I know. And Conway puts a lot of thought into his names, so I think it's best to keep it. I think it's meant to fit into a larger metaphoric …

Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large, $$L(1, …

This is a common point of confusion, and the OP is on exactly the right track. A good reference for the representation theory is Chapter 1, Section 6, of Jacquet-Langlands book "Automorphic forms on …

I think that modular forms (for $SL_2$) of integer and half-integer weights are most important for arithmetic, while modular forms of other (real or complex) weights are primarily objects of analytic …

Upon David Loeffler's request, here is a more fleshed out version of my former comments: In his comment, Nick Ramsey mentioned that the natural L-function for a half-integral weight modular form is re …

I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least c …