What the study found
Under the generalized Riemann hypothesis, the paper gives conditional estimates for the argument of Dirichlet L-functions with large prime modulus. It also gives alternative proofs of several results about 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 indicate that these estimates help in studying low-lying zeros, which are zeros near the central point of an L-function. They also state that their results include a new lower bound on the proportion of Dirichlet L-functions with zeros close to that point.
What the researchers tested
The study works under the generalized Riemann hypothesis, a conjecture in number theory about the location of zeros of L-functions. The author uses Beurling–Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet L-functions for a large prime modulus.
What worked and what didn't
The method produced bounds on the mean and mean square of the argument of Dirichlet L-functions in the stated setting. 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. The abstract does not describe any failed approach or negative result.
What to keep in mind
The results are conditional on the generalized Riemann hypothesis. The abstract also limits the setting to Dirichlet L-functions with a large prime modulus, so the summary does not indicate how far the conclusions extend beyond that case.
Key points
- The paper gives conditional estimates for the argument of Dirichlet L-functions.
- It assumes the generalized Riemann hypothesis.
- Beurling–Selberg extremal functions are used to bound the mean and mean square of the argument.
- The author gives alternative proofs of several results on low-lying zeros.
- The paper reports a new lower bound on the proportion of Dirichlet L-functions with zeros close to the central point.
Disclosure
- Research title:
- Conditional bounds on Dirichlet L-function arguments
- Authors:
- Tianyu Zhao
- Institutions:
- The Ohio State University
- Publication date:
- 2026-04-21
- OpenAlex record:
- View
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