---
title: "Maass waveforms and low-lying zeros"
authors:
  - "Levent Alpoge"
  - "Nadine Amersi"
  - "Geoffrey Iyer"
  - "Oleg Lazarev"
  - "Steven J. Miller"
  - "Liyang Zhang"
year: 2015
venue: "Analytic Number Theory (Maier Festschrift), Springer"
arxiv: "1306.5886"
doi: "10.1007/978-3-319-22240-0_2"
source: "https://arxiv.org/abs/1306.5886"
topics:
  - "number-theory"
  - "modular-forms"
seed_rank: 1216
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 8
lineage: arithmetic-geometry
cites:
  - title: "Low-lying zeroes of Maass form L-functions"
    url: "https://arxiv.org/abs/1301.5702"
    year: 2013
    arxiv: "1301.5702"
    doi: "10.1093/imrn/rnu012"
see:
  - "1215-low-lying-zeroes-of-maass-form-l-functions"
---

# Maass waveforms and low-lying zeros

## One-sentence takeaway

Katz-Sarnak for families of cuspidal Maass forms: orthogonal symmetry, support (-3/2,3/2) as square-free level goes to infinity, and 2-level density to pin the type when level is fixed.

## Why it's lovely

REU-era paper with Miller; the level-aspect companion of 1215.

## Problem

ILS did holomorphic forms; do Maass forms match orthogonal?

## Design

Weighted Kuznetsov / Petersson-style averages; 1- and 2-level densities.

## Evidence

In Pomerance-Rassias (eds.), Analytic Number Theory, Springer 2015, 19-56. DOI 10.1007/978-3-319-22240-0_2. arXiv:1306.5886.

## Limitations

Restricted test functions. Book chapter, not a journal.

## Implications for Broadside

See 1215; do not collapse the two.

## Bottom line

Mint the Springer chapter.

## Links

- arXiv: [1306.5886](https://arxiv.org/abs/1306.5886)
- DOI: [10.1007/978-3-319-22240-0_2](https://doi.org/10.1007/978-3-319-22240-0_2)