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Primes in the intersection of two Piatetski-Shapiro sets
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Primes in the intersection of two Piatetski-Shapiro sets
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Let $\pi(x;\gamma_1,\gamma_2)$ denote the number of primes $p$ with $p\leqslant x$ and $p=\lfloor n^{1/\gamma_1}_1\rfloor=\lfloor n^{1/\gamma_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in\mathbb{R}$ and $1/2<\gamma_2<\gamma_1<1$ are fixed constants. In this paper, we show that $\pi(x;\gamma_1,\gamma_2)$ holds an asymptotic formula for $21/11<\gamma_1+\gamma_2<2$, which constitutes an improvement upon the previous result of Baker [1].
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Cited by 1 Pith paper
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On the distribution of $\alpha p^2$ modulo one in the intersection of two Piatetski--Shapiro sets
For 27/14 < γ1 + γ2 < 2 with 1/2 < γ2 < γ1 < 1, infinitely many primes p in the intersection of two Piatetski-Shapiro sets satisfy ||α p² + β|| < p to the power -(14(γ1+γ2)-27)/43 + ε.
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