On the expected number of roots of a random Dirichlet polynomial
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| Format: | Preprint |
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2024
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| _version_ | 1866913803595153408 |
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| author | Aymone, Marco Bueno, Caio |
| author_facet | Aymone, Marco Bueno, Caio |
| contents | Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of $S_T(t)$ in the dyadic interval $[T,2T]$, say $\mathbb{E} N(T)$, is approximately $2/\sqrt{3}$ times the number of zeros of the Riemann $ζ$ function in the critical strip up to height $T$. Moreover, we also compute the expected number of zeros in the same dyadic interval of the $k$-th derivative of $S_T(t)$. Our proof requires the best upper bounds for the Riemann $ζ$ function known up to date, and also estimates for the $L^2$ averages of certain Dirichlet polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_07375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the expected number of roots of a random Dirichlet polynomial Aymone, Marco Bueno, Caio Number Theory Probability Let $T>0$ and consider the random Dirichlet polynomial $S_T(t)=Re\, \sum_{n\leq T} X_n n^{-1/2-it}$, where $(X_n)_{n}$ are i.i.d. Gaussian random variables with mean $0$ and variance $1$. We prove that the expected number of roots of $S_T(t)$ in the dyadic interval $[T,2T]$, say $\mathbb{E} N(T)$, is approximately $2/\sqrt{3}$ times the number of zeros of the Riemann $ζ$ function in the critical strip up to height $T$. Moreover, we also compute the expected number of zeros in the same dyadic interval of the $k$-th derivative of $S_T(t)$. Our proof requires the best upper bounds for the Riemann $ζ$ function known up to date, and also estimates for the $L^2$ averages of certain Dirichlet polynomials. |
| title | On the expected number of roots of a random Dirichlet polynomial |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2401.07375 |