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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.00660 |
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| _version_ | 1866914230393896960 |
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| author | Guo, Chengliang |
| author_facet | Guo, Chengliang |
| contents | In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting non-equidistribution phenomenon over the full fundamental domain. In fact, this independent equidistribution should be true in a compact set. We further investigate this behaviour by examining a truncated version involving truncated Eisenstein series. Additionally, we propose a conjecture on the joint value distribution of Eisenstein series. The proofs are based on the bounds of the shifted mixed moments of $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00660 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions Guo, Chengliang Number Theory In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we establish asymptotic formulas for these moments. Our results give an interesting non-equidistribution phenomenon over the full fundamental domain. In fact, this independent equidistribution should be true in a compact set. We further investigate this behaviour by examining a truncated version involving truncated Eisenstein series. Additionally, we propose a conjecture on the joint value distribution of Eisenstein series. The proofs are based on the bounds of the shifted mixed moments of $L$-functions. |
| title | Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.00660 |