On Erdős sums of almost primes
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866929607802880000 |
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| author | Gorodetsky, Ofir Lichtman, Jared Duker Wong, Mo Dick |
| author_facet | Gorodetsky, Ofir Lichtman, Jared Duker Wong, Mo Dick |
| contents | In 1935, Erdős proved that the sums $f_k=\sum_n 1/(n\log n)$, over integers $n$ with exactly $k$ prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that $f_k$ is maximized by the prime sum $f_1=\sum_p 1/(p\log p)$. According to a 2013 conjecture of Banks and Martin, the sums $f_k$ are predicted to decrease monotonically in $k$. In this article, we show that the sums restricted to odd integers are indeed monotonically decreasing in $k$, sufficiently large. By contrast, contrary to the conjecture we prove that the sums $f_k$ increase monotonically in $k$, sufficiently large. Our main result gives an asymptotic for $f_k$ which identifies the (negative) secondary term, namely $f_k = 1 - (a+o(1))k^2/2^k$ for an explicit constant $a= 0.0656\cdots$. This is proven by a refined method combining real and complex analysis, whereas the classical results of Sathe and Selberg on products of $k$ primes imply the weaker estimate $f_k=1+O_{\varepsilon}(k^{\varepsilon-1/2})$. We also give an alternate, probability-theoretic argument related to the Dickman distribution. Here the proof reduces to showing a sequence of integrals converges exponentially quickly to $e^{-γ}$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08277 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Erdős sums of almost primes Gorodetsky, Ofir Lichtman, Jared Duker Wong, Mo Dick Number Theory Probability 11N25, 11Y60, 11A05, 60G18, 60H25 In 1935, Erdős proved that the sums $f_k=\sum_n 1/(n\log n)$, over integers $n$ with exactly $k$ prime factors, are bounded by an absolute constant, and in 1993 Zhang proved that $f_k$ is maximized by the prime sum $f_1=\sum_p 1/(p\log p)$. According to a 2013 conjecture of Banks and Martin, the sums $f_k$ are predicted to decrease monotonically in $k$. In this article, we show that the sums restricted to odd integers are indeed monotonically decreasing in $k$, sufficiently large. By contrast, contrary to the conjecture we prove that the sums $f_k$ increase monotonically in $k$, sufficiently large. Our main result gives an asymptotic for $f_k$ which identifies the (negative) secondary term, namely $f_k = 1 - (a+o(1))k^2/2^k$ for an explicit constant $a= 0.0656\cdots$. This is proven by a refined method combining real and complex analysis, whereas the classical results of Sathe and Selberg on products of $k$ primes imply the weaker estimate $f_k=1+O_{\varepsilon}(k^{\varepsilon-1/2})$. We also give an alternate, probability-theoretic argument related to the Dickman distribution. Here the proof reduces to showing a sequence of integrals converges exponentially quickly to $e^{-γ}$, which may be of independent interest. |
| title | On Erdős sums of almost primes |
| topic | Number Theory Probability 11N25, 11Y60, 11A05, 60G18, 60H25 |
| url | https://arxiv.org/abs/2303.08277 |