Mean values of ratios of the Riemann zeta function
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866911891415105536 |
|---|---|
| author | Yang, Daodao |
| author_facet | Yang, Daodao |
| contents | It is proved that
$$\int_{T}^{2T} \left|\frac{ζ\left(\frac{1}{2}+{\rm i} t\right)}{ζ\left(1+2{\rm i} t\right)}\right|^2 {\rm d} t = \frac{1}{ζ(2)} T \log T + \left( \frac{\log \frac{2}π + 2γ-1 }{ζ(2)} -4 \,\frac{ζ^{\prime}(2)}{ζ^2(2)} \right) T + O\left(T\, \left(\log T\right)^{-2023} \right) , \quad \forall T \geqslant 100. $$
For given $a \in \mathbb N$, we also establish similar formulas for second moments of $|ζ(\tfrac{1}{2} + {\rm i} t)/ζ(1 + {\rm i} at)|.$ We have \begin{align*}
\lim_{a \to \infty} \lim_{T \to \infty}\frac{1}{T \log T} \int_{T}^{2T} \left|\frac{ζ\left(\frac{1}{2}+{\rm i} t\right)}{ζ\left(1+{\rm i} at\right)}\right|^2 {\rm d} t = \frac{ζ(2)}{ζ(4)}.
\end{align*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08091 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mean values of ratios of the Riemann zeta function Yang, Daodao Number Theory It is proved that $$\int_{T}^{2T} \left|\frac{ζ\left(\frac{1}{2}+{\rm i} t\right)}{ζ\left(1+2{\rm i} t\right)}\right|^2 {\rm d} t = \frac{1}{ζ(2)} T \log T + \left( \frac{\log \frac{2}π + 2γ-1 }{ζ(2)} -4 \,\frac{ζ^{\prime}(2)}{ζ^2(2)} \right) T + O\left(T\, \left(\log T\right)^{-2023} \right) , \quad \forall T \geqslant 100. $$ For given $a \in \mathbb N$, we also establish similar formulas for second moments of $|ζ(\tfrac{1}{2} + {\rm i} t)/ζ(1 + {\rm i} at)|.$ We have \begin{align*} \lim_{a \to \infty} \lim_{T \to \infty}\frac{1}{T \log T} \int_{T}^{2T} \left|\frac{ζ\left(\frac{1}{2}+{\rm i} t\right)}{ζ\left(1+{\rm i} at\right)}\right|^2 {\rm d} t = \frac{ζ(2)}{ζ(4)}. \end{align*} |
| title | Mean values of ratios of the Riemann zeta function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2307.08091 |