What the study found
Under the generalized Riemann hypothesis, the study gives bounds for the mean and mean square of the argument of Dirichlet L-functions for a large prime modulus. It also reports applications to low-lying zeros, including a new lower bound on the proportion of Dirichlet L-functions with zeros close to the central point.
Why the authors say this matters
The authors use these bounds to give alternative proofs of several results on low-lying zeros of Dirichlet L-functions. The study suggests this approach also yields a new lower bound on how many such functions have zeros near the central point.
What the researchers tested
The researchers worked under the generalized Riemann hypothesis and used Beurling-Selberg extremal functions, a tool for constructing sharp upper and lower bounds. They applied this to the argument of Dirichlet L-functions for a large prime modulus.
What worked and what didn't
The method produced bounds on both the mean and mean square of the argument of Dirichlet L-functions. It also yielded alternative proofs of several low-lying zero results and a new lower bound on the proportion of Dirichlet L-functions with zeros near the central point. In particular, the authors show conditionally that for any fixed value, there is a positive proportion of Dirichlet L-functions whose first zero lies below that value times the average spacing between consecutive zeros.
What to keep in mind
The results are conditional on the generalized Riemann hypothesis. The abstract does not give the full range of the modulus, the exact constants, or further limitations beyond this condition.
Key points
- The study gives conditional bounds for the mean and mean square of the argument of Dirichlet L-functions.
- It uses Beurling-Selberg extremal functions and assumes the generalized Riemann hypothesis.
- The results are applied to low-lying zeros of Dirichlet L-functions.
- The authors obtain a new lower bound on the proportion of functions with zeros close to the central point.
- They show conditionally that a positive proportion have first zeros below a fixed multiple of the average zero spacing.
Disclosure
- Research title:
- Conditional bounds on Dirichlet function arguments and low-lying zeros
- Authors:
- Tianyu Zhao
- Institutions:
- The Ohio State University
- Publication date:
- 2026-04-21
- DOI:
- 10.1112/blms.70360
- OpenAlex record:
- View
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