On Erdős sums of almost primes
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 s...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.650/ |
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| author | Gorodetsky, Ofir Lichtman, Jared Duker Wong, Mo Dick |
| author_facet | Gorodetsky, Ofir Lichtman, Jared Duker Wong, Mo Dick |
| author_sort | Gorodetsky, Ofir |
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| description | 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 $\mathrm{e}^{-\gamma }$, which may be of independent interest. |
| format | Article |
| id | doaj-art-08e4bdc305b8413db491aded4972df72 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-08e4bdc305b8413db491aded4972df722025-02-07T11:26:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G121571159610.5802/crmath.65010.5802/crmath.650On Erdős sums of almost primesGorodetsky, Ofir0Lichtman, Jared Duker1Wong, Mo Dick2Department of Mathematics, Technion – Israel Institute of Technology, Haifa 3200003, Israel; Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UKDepartment of Mathematics, Stanford University, Stanford, CA, USADepartment of Mathematical Sciences, Durham University, Stockton Road, Durham DH1 3LE, UKIn 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 $\mathrm{e}^{-\gamma }$, which may be of independent interest.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.650/Almost primesprimitive setDickman distributionrecursive distributional equation |
| spellingShingle | Gorodetsky, Ofir Lichtman, Jared Duker Wong, Mo Dick On Erdős sums of almost primes Comptes Rendus. Mathématique Almost primes primitive set Dickman distribution recursive distributional equation |
| title | On Erdős sums of almost primes |
| title_full | On Erdős sums of almost primes |
| title_fullStr | On Erdős sums of almost primes |
| title_full_unstemmed | On Erdős sums of almost primes |
| title_short | On Erdős sums of almost primes |
| title_sort | on erdos sums of almost primes |
| topic | Almost primes primitive set Dickman distribution recursive distributional equation |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.650/ |
| work_keys_str_mv | AT gorodetskyofir onerdossumsofalmostprimes AT lichtmanjaredduker onerdossumsofalmostprimes AT wongmodick onerdossumsofalmostprimes |