Number Theory Commons^™

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

University Honors Theses

This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

An Introduction To Number Theory, J. J. P. Veerman

PDXOpen: Open Educational Resources

These notes are intended for a graduate course in Number Theory. No prior familiarity with number theory is assumed.

Chapters 1-14 represent almost 3 trimesters of the course. Eventually we intend to publish a full year (3 trimesters) course on number theory. The current content represents courses the author taught in the academic years 2020-2021 and 2021-2022.

It is a work in progress. If you have questions or comments, please contact Peter Veerman (veerman@pdx.edu).

On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, Liubomir Chiriac

Mathematics and Statistics Faculty Publications and Presentations

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two.