Conditional bounds for Dirichlet function arguments and low-lying zeros

What the study found: Under the generalized Riemann hypothesis, the paper gives conditional estimates for the argument of Dirichlet L-functions when the modulus is a large prime. It also reports applications to low-lying zeros and a new lower bound for the proportion of Dirichlet L-functions with zeros close to the central point.
Why the authors say this matters: The authors say these estimates lead to alternative proofs of several results about low-lying zeros, and they conclude that their new lower bound adds information about how often zeros occur near the central point. In this context, low-lying zeros are zeros of L-functions close to the point where the function is centered.
What the researchers tested: The study uses Beurling–Selberg extremal functions, which are special approximating functions used in analysis, to bound the mean and mean square of the argument of Dirichlet L-functions for a large prime modulus. The work is conditional on the generalized Riemann hypothesis.
What worked and what didn't: The authors report that they could bound the mean and mean square of the argument in the stated setting. They also say they obtained alternative proofs of several results on low-lying zeros and a new lower bound on the proportion of Dirichlet L-functions with zeros close to the central point; the abstract does not state any failures or negative results.
What to keep in mind: The results are conditional on the generalized Riemann hypothesis, and the abstract restricts the setting to a large prime modulus. No further limitations are described in the available summary.

Key points

• The paper gives conditional estimates for the argument of Dirichlet L-functions.
• The results assume the generalized Riemann hypothesis and a large prime modulus.
• Beurling–Selberg extremal functions are used to bound the mean and mean square of the argument.
• The authors say the work yields alternative proofs of several results on low-lying zeros.
• The abstract states a new lower bound for the proportion of Dirichlet L-functions with zeros close to the central point.