Moments of real Dirichlet $L$-functions and multiple Dirichlet series
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909144268668928 |
|---|---|
| author | Čech, Martin |
| author_facet | Čech, Martin |
| contents | We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the generalized Lindelöf hypothesis for $k\geq 5$.
As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any $k$ (conditionally under the Generalized Lindelöf Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials.
The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moments of real Dirichlet $L$-functions and multiple Dirichlet series Čech, Martin Number Theory 11M06 (primary), 11M32 (secondary) We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the generalized Lindelöf hypothesis for $k\geq 5$. As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any $k$ (conditionally under the Generalized Lindelöf Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials. The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith. |
| title | Moments of real Dirichlet $L$-functions and multiple Dirichlet series |
| topic | Number Theory 11M06 (primary), 11M32 (secondary) |
| url | https://arxiv.org/abs/2402.07473 |