• obeckwith

Modular forms and L-functions

Updated: Mar 11, 2021

The study of L-functions such as the Riemann zeta function is one of the most classical fields of analytic number theory. The distribution of zeros of L-functions encode arithmetic information such as the distribution of primes and coefficients of modular forms. For example, the Generalized Riemann Hypothesis predicts that all the nontrivial zeros of L-functions have real part equal to ½, and knowing this for the Riemann zeta function would give us a very strong approximation for the prime counting function. With Liu, Thorner, and Zaharescu, I have been studying statistical approximations to the Generalized Riemann Hypothesis. As an application, we've explored the distributions of the fractional parts of the zeros of a class of L-functions. We would be able to make several of our applications more precise if we could quantify the error terms in some of our estimates, but this involves a certain Diophantine approximation question that does not seem straightforward, and it is something I hope to make progress on in the future.

1 view0 comments

Recent Posts

See All

Imagine you have a sphere in R^3 with radius equal to the square root of n. How many points with integer coordinates are there on this sphere? Gauss gave a formula in Disquisitiones Arithmeticae relat

Say you want to know how many ways you can write n as a sum of positive integers, up to reordering the summands. The number of ways is the nth partition number p(n). 100 years ago, Ramanujan discove

I work in some of the more abstract parts of mathematics, but I also seek out situations where the theoretical ideas can be put to good use in the sciences. As a graduate student, I wrote a paper with