My preprints can all be found through my arXiv identifier [![[[email protected]|24x24]]](https://arxiv.org/a/williams_c_4.html) or on Google Scholar [![[scholar_icon.png|24x24]]](https://scholar.google.com/citations?user=LRWv7xkAAAAJ&hl=en&oi=sra). *** 3\. On the Pair Correlation of Zeros of L-Functions for Non-CM Newforms in Shifted Ranges, with [Di Liu](https://math.illinois.edu/directory/profile/dil4) and [Alexandru Zaharescu](https://math.illinois.edu/directory/profile/zaharesc). Submitted, 2024. >[!note]- Summary >We introduce a new degree 4 $L$-function $L_\lambda$ built from shifted copies of an $L$-function for newforms, and prove unconditional results about pair correlations between the zeros of this $L_\lambda$. In particular, the distribution of zeros of this new $L_\lambda$ function have implications for gaps of constant size between zeros of the original newform $L$-function, even at very large heights. 2\. Hecke Relations for Eta Multipliers and Congruences of Crank and Rank Moments. [*Journal of Number Theory*](https://doi.org/10.1016/j.jnt.2025.03.006), 2025. >[!note]- Summary >I prove explicit families of identities for Eisenstein-Eta quotients $\frac{E_k}{\eta^s}$ from Hecke operators acting on the spaces $M_{k-s/2}(\operatorname{SL}_2(\mathbb{Z}))$, and use these identities to prove new systematic congruences for modular forms and partition statistics modulo arbitrary powers of primes. This paper resolves a previously open question of [Frank Garvan](https://www.qseries.org/fgarvan/)'s for the higher-order smallest parts functions $\operatorname{spt}_{2,3,4,5}$. 1\. An Infinite Family of Vector-Valued Mock Theta Functions, with [Nick Andersen](https://mathdept.byu.edu/~nick/). [*The Ramanujan Journal*](https://doi.org/10.1007/s11139-023-00745-x), 2023. >[!note]- Summary >Ramanujan's mock theta functions are beautiful, enigmatic objects that have fascinated mathematicians for more than 100 years. We take an infinite number of families of recently discovered examples of mock theta functions --- parameterized by a value called the *order* --- and assign to each family an explicit vector-valued weak harmonic Maaß form transforming according to the Weil representation. > $\qquad$ 1\. a. Vector-Valued Mock Theta Functions. [Master's Thesis, BYU Department of Mathematics](https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=10657&context=etd), 2022. (This result is greatly strengthened and simplified in 1 above).  0\. Magnetic Field Modulation Toward Particle Accelerator RF Source Replacement. [*Journal of the Utah Academy of Sciences, Arts, and Letters*](https://www.utahacademy.org/wp-content/uploads/2020/05/2019-JUASAL-full-text-final.pdf), page 249 ff, 2019. >[!note]- Summary >This reports on research I completed under the tutelage of [Haipeng Wang](https://www.linkedin.com/in/haipeng-wang-abb2797a/) at [Jefferson Lab](https://education.jlab.org/ugresearch/18-21.html). Particle accelerators require a source of radio waves to drive their superconducting cavities. Currently, this radio frequency energy is usually provided by klystrons, which are not as power-efficient as magnetrons but have much cleaner signals and controls. Because of this it would be desirable to control the frequency and noise of microwave magnetrons for use in particle accelerators. In this paper we reported on: >1. Optimization of a trim coil and its bipolar power supply for magnetic field modulation; >2. Driving of a warm Niobium cavity with a 2.45-GHz magnetron; >3. Design of a 1497-MHz magnetron test stand. ![[sleepy.png|250]]