Bandwidth of #coprimes to p# in intervals of fixed length

mart_r · 2021-03-27, 17:37

At the moment I'm interested in the behavior of numbers coprime to p#, or p-quasiprimes, as they are also called.

Specifically, a question regarding the bandwidth of the number of coprimes that can appear in a fixed interval of length x.
Example: for p=5, the maximal bandwidth is 4: for 8 <x<10 there's a minimum of 1 and a maximum of 4 coprimes, and for 12<x<14, there's a minimum of 2 and a maximum of 5 coprimes to 5#.

I'd be grateful for any hint leading to papers that touch this particular subject.

mart_r · 2021-03-28, 17:55

Well, maybe that was too specific.

BTW, I already went through some works of Maier, Cadwell, and related works of Granville, Pintz and some others, and was hoping I overlooked some lesser-known papers...

Maybe here's an easier question. I still have some serious deficiencies in dealing with differential equations... Is there a readily available algorithm, preferably for Pari, that calculates values for the differential delay equation of Buchstab? You know, the omega(u) one from this Cheer-Goldston paper:
https://www.ams.org/journals/mcom/19...990-1023043-8/

2021-03-27, 17:37	#1
mart_r Dec 2008 you know...around... 2³·109 Posts	Bandwidth of #coprimes to p# in intervals of fixed length At the moment I'm interested in the behavior of numbers coprime to p#, or p-quasiprimes, as they are also called. Specifically, a question regarding the bandwidth of the number of coprimes that can appear in a fixed interval of length x. Example: for p=5, the maximal bandwidth is 4: for 8 <x<10 there's a minimum of 1 and a maximum of 4 coprimes, and for 12<x<14, there's a minimum of 2 and a maximum of 5 coprimes to 5#. I'd be grateful for any hint leading to papers that touch this particular subject.

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2021-03-28, 17:55	#2
mart_r Dec 2008 you know...around... 1101101000₂ Posts	Well, maybe that was too specific. BTW, I already went through some works of Maier, Cadwell, and related works of Granville, Pintz and some others, and was hoping I overlooked some lesser-known papers... Maybe here's an easier question. I still have some serious deficiencies in dealing with differential equations... Is there a readily available algorithm, preferably for Pari, that calculates values for the differential delay equation of Buchstab? You know, the omega(u) one from this Cheer-Goldston paper: https://www.ams.org/journals/mcom/19...990-1023043-8/