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Francesco Polizzi
Francesco Polizzi
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For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: all\ prime\ factors\ of\ q \in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q) \end{equation}\begin{equation} |\{q\in \mathbb{N},q<Q: \text{all prime factors of } q\,\in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q) \end{equation} where $d(n,Q):=|\{m\in \mathbb{N},m<Q: m\mid n\}|$.

This lemma appears below (3.46) in Bourgain's article "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations". I'm not familiar with number theory; could you please explain how to deduce this conclusion and give some references on number theory?

I'm not familiar with number theory; could you please explain how to deduce this conclusion and give some references on number theory?

For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: all\ prime\ factors\ of\ q \in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q) \end{equation} where $d(n,Q):=|\{m\in \mathbb{N},m<Q: m\mid n\}|$.

This lemma appears below (3.46) in Bourgain's article "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations". I'm not familiar with number theory; could you please explain how to deduce this conclusion and give some references on number theory?

For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: \text{all prime factors of } q\,\in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q) \end{equation} where $d(n,Q):=|\{m\in \mathbb{N},m<Q: m\mid n\}|$.

This lemma appears below (3.46) in Bourgain's article "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations".

I'm not familiar with number theory; could you please explain how to deduce this conclusion and give some references on number theory?

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Dapao Zhang
Dapao Zhang
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Estimates about prime numbers: a lemma in Bourgain's article

For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: all\ prime\ factors\ of\ q \in A\}|<Ce^{c\frac{\log Q}{\log \log Q}}d(n,Q) \end{equation} where $d(n,Q):=|\{m\in \mathbb{N},m<Q: m\mid n\}|$.

This lemma appears below (3.46) in Bourgain's article "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations". I'm not familiar with number theory; could you please explain how to deduce this conclusion and give some references on number theory?