What the study found
Under the generalized Riemann hypothesis, the study gives conditional bounds on the mean and mean square of the argument of Dirichlet L-functions for large prime modulus. It also reports applications to low-lying zeros and 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 say these bounds have applications to results about low-lying zeros, which are zeros of an L-function near the central point. They also conclude that their result gives a new lower bound on how many Dirichlet L-functions have zeros close to that point.
What the researchers tested
The paper works conditionally under the generalized Riemann hypothesis, a widely studied conjecture about the zeros of L-functions. The authors use Beurling–Selberg extremal functions to estimate the mean and mean square of the argument of Dirichlet L-functions for a large prime modulus.
What worked and what didn't
The conditional estimates were used to give alternative proofs of several results on low-lying zeros of Dirichlet L-functions. The abstract also states that, for any fixed parameter less than 1, there is a positive proportion of Dirichlet L-functions whose first zero has height less than that parameter 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 provide further limitations, and the exact parameter values are not fully shown in the provided text.
Key points
- The paper proves conditional estimates for the argument of Dirichlet L-functions.
- The estimates cover both the mean and mean square for large prime modulus.
- The authors use Beurling–Selberg extremal functions in their method.
- The results are applied to low-lying zeros of Dirichlet L-functions.
- The abstract states a new lower bound for the proportion of Dirichlet L-functions with zeros near the central point.
Disclosure
- Research title:
- Conditional bounds for Dirichlet L-function arguments and low-lying zeros
- Authors:
- Tianyu Zhao
- Institutions:
- The Ohio State University
- Publication date:
- 2026-04-21
- OpenAlex record:
- View
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