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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2109.03767 |
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| _version_ | 1866913754870972416 |
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| author | Kuperberg, Vivian |
| author_facet | Kuperberg, Vivian |
| contents | Montgomery and Soundararajan showed that the distribution of $ψ(x+H) - ψ(x)$, for $0 \le x \le N$, is approximately normal with mean $ \sim H$ and variance $\sim H \log (N/H)$, when $N^δ \le H \le N^{1-δ}$. Their work depends on showing that sums $R_k(h)$ of $k$-term singular series are $μ_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, where $A$ is a constant and $μ_k$ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when $k$ is odd, $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$. We prove an upper bound with the correct power of $h$ when $k = 3$, and prove analogous upper bounds in the function field setting when $k =3$ and $k = 5$. We provide further evidence for this conjecture in the form of numerical computations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_03767 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Odd moments in the distribution of primes Kuperberg, Vivian Number Theory Montgomery and Soundararajan showed that the distribution of $ψ(x+H) - ψ(x)$, for $0 \le x \le N$, is approximately normal with mean $ \sim H$ and variance $\sim H \log (N/H)$, when $N^δ \le H \le N^{1-δ}$. Their work depends on showing that sums $R_k(h)$ of $k$-term singular series are $μ_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, where $A$ is a constant and $μ_k$ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when $k$ is odd, $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$. We prove an upper bound with the correct power of $h$ when $k = 3$, and prove analogous upper bounds in the function field setting when $k =3$ and $k = 5$. We provide further evidence for this conjecture in the form of numerical computations. |
| title | Odd moments in the distribution of primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2109.03767 |