On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918201021956096 |
|---|---|
| author | Kim, Jiseong Nath, Kunjakanan |
| author_facet | Kim, Jiseong Nath, Kunjakanan |
| contents | Let $\{λ_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, χ)$, where $χ$ is a Dirichlet character modulo $q$, that if $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ with $\varepsilon>0$, then for any $A\geq 1$, $$\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A} \frac{HX^2}{(\log X)^{A}}$$ holds. Moreover, under an additional hypothesis on the fourth moment of certain Dirichlet polynomials (which follows from GRH for $L(s, f)$), we show that the above result can be strengthened to hold in a wider range $X^{1/3+\varepsilon}\ll H \ll X^{1-\varepsilon}$. Finally, if we average over the forms $f$, then for $X^{\varepsilon}\ll H\ll X^{1-\varepsilon}$ and for any $A\geq 1$,
$$ \sum_{f\in \mathcal{H}_k}ω_f\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A}\frac{HX^2}{(\log X)^{A}},$$ where $\mathcal{H}_k$ is the Hecke basis for the space of holomorphic cusp forms of weight $k$ for the full modular group $\mathrm{SL}(2, \mathbb{Z})$ and $ω_f$ are harmonic weights associated with $f\in \mathcal{H}_k$. These results may be viewed as modular analogues of the averaged forms of the Hardy--Littlewood prime tuple conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10594 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments Kim, Jiseong Nath, Kunjakanan Number Theory 11F30, 11N37, 11P55 Let $\{λ_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, χ)$, where $χ$ is a Dirichlet character modulo $q$, that if $X^{2/3+\varepsilon} \ll H \ll X^{1-\varepsilon}$ with $\varepsilon>0$, then for any $A\geq 1$, $$\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A} \frac{HX^2}{(\log X)^{A}}$$ holds. Moreover, under an additional hypothesis on the fourth moment of certain Dirichlet polynomials (which follows from GRH for $L(s, f)$), we show that the above result can be strengthened to hold in a wider range $X^{1/3+\varepsilon}\ll H \ll X^{1-\varepsilon}$. Finally, if we average over the forms $f$, then for $X^{\varepsilon}\ll H\ll X^{1-\varepsilon}$ and for any $A\geq 1$, $$ \sum_{f\in \mathcal{H}_k}ω_f\sum_{1\leq |h|\leq H}\bigg| \sum_{\substack{X<n,\: m \leq 2X \\ n - m = h}} λ_f(n)Λ(n)λ_f(m)Λ(m) \bigg|^2 \ll_{A}\frac{HX^2}{(\log X)^{A}},$$ where $\mathcal{H}_k$ is the Hecke basis for the space of holomorphic cusp forms of weight $k$ for the full modular group $\mathrm{SL}(2, \mathbb{Z})$ and $ω_f$ are harmonic weights associated with $f\in \mathcal{H}_k$. These results may be viewed as modular analogues of the averaged forms of the Hardy--Littlewood prime tuple conjecture. |
| title | On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments |
| topic | Number Theory 11F30, 11N37, 11P55 |
| url | https://arxiv.org/abs/2511.10594 |