The Prime Geodesic Theorem and Bounds for Character Sums
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917627275771904 |
|---|---|
| author | Kaneko, Ikuya |
| author_facet | Kaneko, Ikuya |
| contents | We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known conditionally on the generalised Lindelöf hypothesis for quadratic Dirichlet $L$-functions. Our argument goes through a well-trodden trail via the automorphic machinery, and refines the techniques of Iwaniec (1984) and Cai (2002) to a maximum extent. A key ingredient is an asymptotic for bilinear forms with a counting function in Kloosterman sums via hybrid Weyl-strength subconvex bounds for quadratic Dirichlet $L$-functions due to Young (2017), zero density estimates due to Heath-Brown (1995) near the edge of the critical strip, and an asymptotic for averages of Zagier $L$-series due to Balkanova-Frolenkov-Risager (2022). Furthermore, we strengthen our exponent to $\frac{5}{8}+\varepsilon$ conditionally on the generalised Lindelöf hypothesis for quadratic Dirichlet $L$-functions, which breaks the existing barrier. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Prime Geodesic Theorem and Bounds for Character Sums Kaneko, Ikuya Number Theory 11L40 (primary), 11F72, 11M26 (secondary) We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known conditionally on the generalised Lindelöf hypothesis for quadratic Dirichlet $L$-functions. Our argument goes through a well-trodden trail via the automorphic machinery, and refines the techniques of Iwaniec (1984) and Cai (2002) to a maximum extent. A key ingredient is an asymptotic for bilinear forms with a counting function in Kloosterman sums via hybrid Weyl-strength subconvex bounds for quadratic Dirichlet $L$-functions due to Young (2017), zero density estimates due to Heath-Brown (1995) near the edge of the critical strip, and an asymptotic for averages of Zagier $L$-series due to Balkanova-Frolenkov-Risager (2022). Furthermore, we strengthen our exponent to $\frac{5}{8}+\varepsilon$ conditionally on the generalised Lindelöf hypothesis for quadratic Dirichlet $L$-functions, which breaks the existing barrier. |
| title | The Prime Geodesic Theorem and Bounds for Character Sums |
| topic | Number Theory 11L40 (primary), 11F72, 11M26 (secondary) |
| url | https://arxiv.org/abs/2402.12133 |