On the Dirichlet L-functions and the L-functions of Cusp Forms
Talk, Bowdoin College, Department of Mathematics, Brunswick, Maine
Upon my graduation from Bowdoin College in Maine, I have presented my honors thesis to the mathematics department. The topic is mainly about two families of \(L\)-functions, namely, the Dirichlet \(L\)-functions and the \(L\)-functions of cups forms. The first family is where the Riemann zeta function belong to while the second one is more special as we have to acquire some knowledge about modular forms to understand it.
This work was a spin-off from the paper Walking to Infinity Along Some Number Theory sequences. However, instead of observing sequences of integers which are one-dimensional, we consider primes in the real quadratic integer rings in \(\mathbb{Z}[\sqrt{2}]\), which are of the form \(a+b\sqrt{2}\). This study was inspired by the Gaussian Moat problem posed by Basil Gordon in 1962, and the work has been presented in a couple of conferences by my collegue, Bencheng Li and Daniel Sarnecki.
This work was presented at the PaJAMAS (The PAlmetto Joint Arithmetic, Modularity, and Analysis Serie) conference 2020 by Tudor Popescu, Saam Rasool, and me, but it is a joint work with Prefessor Steven J. Miller, Joshua M. Siktar, and PolymathREU Prime Walk group in summer 2020. The paper is under review, so I can only provide the link to the old version of this paper which is on ArXiv.
This was the first talk of the time I was at Bowdoin. The project was done during my sophomore summer under the supervision of Professor Naomi Tanabe. The poster presentation itself happened in October of my Junior year. However, at that time, we had not found the integral representation of the Dirichlet \(\beta\)-function evaluated at even integers. The work was finally completed and submitted in May 2021 and currently under review.
In 2015, Poom Lertpinyowong, Krittamed Lengrugsa, and I contucted the math project together. The project has been presented in Thaiand’s YSC (Young Scientist Competition) 2015 and the Kolmokorov readings in 2016. The advisors of this work were Professors Sittichoke Som-am from Mahidol Wittayanusorn School and Wacharin Wichiramala from Chulalongkorn University.