Prime numbers of the form \([n^2]\) (Q2731117)
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scientific article; zbMATH DE number 1625563
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Prime numbers of the form \([n^2]\) |
scientific article; zbMATH DE number 1625563 |
Statements
21 February 2002
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Piatetski-Shapiro prime number theorem
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sieve methods
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Prime numbers of the form \([n^2]\) (English)
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Let \(\pi_c(x)= |\{n\leq x: [n^c]\) is a prime number\(\}|\), where \(c>1\) and \([t]\) denotes the integral part of \(t\). The authors prove the inequality \(\pi_c(x)\gg x/c\log x\) for \(1< c< 243/205\approx 1.18536\) and \(x\geq x_0(c)\). They use sieve methods combined with estimates of exponential sums. To obtain the desired result, some numerical calculations are necessary which have be done with the aid of Maple.
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