I'm looking for a result (embarrassingly enough, a somewhat famous result) which shows the infinitude in some sense I don't recall of primes of the form $ \lfloor x^k\rfloor $ for $k$ fixed and irrational. There were sharp limits on the size of $k.$ I think the original result has been improved many times, mostly by widening the allowable range of $k.$
Primes of the form $\lfloor x^k\rfloor$
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number-theory
reference-request
prime-numbers
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0The Alkauskas-Dubickas result has varying integer exponents, you're after fixed irrational exponents, so I'd say it's not that similar. – 2012-06-14
1 Answers
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Rivat and Sargos, Nombres premiers de la forme $[n^c]$, Canad. J. Math. 53 (2001), no. 2, 414–433, MR1820915 (2002a:11107), reviewed by G. Greaves.
The authors establish an asymptotic formula for the number of primes not exceeding $x$ of the form $[n^c]$. Their result applies for each $c$ with $1\lt c\lt2817/2426$. The review compares this to previous work, and there are links to other papers and reviews that cite this paper.
Apparently the first paper along these lines was by Piatetski-Shapiro in 1953, with $1\lt c\lt12/11$.
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0That's it. I've heard it called the "Piatetski-Shapiro Prime Number Theorem" because of its density result: $x/c\log x.$ – 2012-06-14