mersenneforum.org Incorrect guess based on limited data: Number of Primes 6k-1 > Number of Primes 6k+1
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2021-08-26, 19:45   #12
Dobri

"刀-比-日"
May 2018

23810 Posts

Quote:
 Originally Posted by charybdis Don't forget 2 is a prime of the form 2 mod 3.
My bad, the Wolfram script is actually testing the π6,5(x) and π6,1(x) cases.

Therefore, the following correction applies: π6,5(x) = π6,1(x) for x = 2, 3, 7, 13, 19, 37, 43, 79, 163, 223 and 229.
π6,5(2) = π6,1(2) = 0 (trivial case)
π6,5(3) = π6,1(3) = 0 (trivial case)
π6,5(7) = π6,1(7) = 1
π6,5(13) = π6,1(13) = 2
π6,5(19) = π6,1(19) = 3
π6,5(37) = π6,1(37) = 5
π6,5(43) = π6,1(43) = 6
π6,5(79) = π6,1(79) = 10
π6,5(163) = π6,1(163) = 18
π6,5(223) = π6,1(223) = 23
π6,5(229) = π6,1(229) = 24

Last fiddled with by Dobri on 2021-08-26 at 20:10

 2021-08-26, 20:14 #13 bhelmes     Mar 2016 32·41 Posts Perhaps this is helpful for you: p 1 3 5 mod 6 ------------------ 10 1 1 1 10^2 11 1 12 10^3 80 1 86 10^4 611 1 616 10^5 4784 1 4806 10^6 39231 1 39265 10^7 332194 1 332383 10^8 2880517 1 2880936 10^9 25422713 1 25424819 10^10 227523123 1 227529386 10^11 2059018668 1 2059036143 10^12 18803933520 1 18803978496 10^13 173032692013 1 173032844824 10^14 1602470745574 1 1602471005226
2021-08-26, 20:31   #14
Dobri

"刀-比-日"
May 2018

EE16 Posts

Quote:
 Originally Posted by bhelmes Perhaps this is helpful for you: p 1 3 5 mod 6 ...
Thanks, this is of limited use as the task is to study the exact locations of the reversal points for which the sign of π6,5(x) - π6,1(x) flips to the opposite, or eventually π6,5(x) - π6,1(x) = 0 with or without a subsequent sign reversal for larger primes. There is no need of considering the only prime π6,3(x) = 3 for said task.

2021-08-26, 22:36   #15
Dobri

"刀-比-日"
May 2018

EE16 Posts

Quote:
 Originally Posted by charybdis Littlewood's result from 1914, quoted in Granville and Martin's survey, shows that the difference oscillates from positive to negative infinitely many times, and also takes arbitrarily large positive and negative values.
Indeed, the question is what is happening on the average (rather than at distinct sampling points or limited sampling intervals) when the number of reversal points approaches infinity.

 2021-08-26, 23:31 #16 Dobri   "刀-比-日" May 2018 2·7·17 Posts It should be noted that the prime number race π6,5(x) vs. π6,1(x) is not the same with the race π3,2(x) vs. π3,1(x) and their respective reversal points might differ. Therefore, unless a reference is found to prove that the reversal points of the race π6,5(x) vs. π6,1(x) have been studied in the past, the said race is a new topic. Thus I do not understand why an anonymous mod had to change the thread icon from 'question' sign to 'minus' sign.
2021-08-27, 00:00   #17
charybdis

Apr 2020

10001000102 Posts

Quote:
 Originally Posted by Dobri Indeed, the question is what is happening on the average (rather than at distinct sampling points or limited sampling intervals) when the number of reversal points approaches infinity.
It's likely unknown what happens to the average value of C in the long term, as we don't even know that the Chebyshev bias exists without assuming GRH. However, I would be surprised if it tended to any actual limit, as the fluctuations are too irregular. It might even be possible to prove this from known results, I'm not an expert here.

Quote:
 Originally Posted by Dobri Therefore, unless a reference is found to prove that the reversal points of the race π6,5(x) vs. π6,1(x) have been studied in the past, the said race is a new topic.
This has been investigated. See OEIS A096449.

2021-08-27, 00:17   #18
Dobri

"刀-比-日"
May 2018

2·7·17 Posts

Quote:
 Originally Posted by charybdis This has been investigated. See OEIS A096449.
The link to OEIS only shows how the mod 3 terms can be rearranged to list the mod 6 terms, it is trivial.
There is no investigation of any kind in OEIS.

 2021-08-27, 00:19 #19 charybdis     Apr 2020 2×3×7×13 Posts What sort of investigation do you want? From a mathematical point of view, there's no real difference between the mod 3 and mod 6 races. Results like Littlewood's and Rubinstein and Sarnak's aren't going to be changed by the omission of the single prime 2.
2021-08-27, 00:25   #20
Dobri

"刀-比-日"
May 2018

23810 Posts

Quote:
 Originally Posted by charybdis What sort of investigation do you want? From a mathematical point of view, there's no real difference between the mod 3 and mod 6 races. Results like Littlewood's and Rubinstein and Sarnak's aren't going to be changed by the omission of the single prime 2.
There appears to be a significant difference. For instance, the list of π6,5(x) = π6,1(x) in my corrected post is invalid for the mod 3 case. Changing the sequence interval from 3 to 6 changes the behavior of the prime-counting function.

 2021-08-27, 00:35 #21 charybdis     Apr 2020 2×3×7×13 Posts You get a few extra crossover points by going from mod 3 to mod 6. That's it. Number theorists are not particularly concerned with the exact positions of the crossover points - especially once the first one has been found - but about the long-term trends. Changing the difference by 1 does not affect these. The difference is usually much larger than 1.
2021-08-27, 00:48   #22
Dobri

"刀-比-日"
May 2018

23810 Posts

Quote:
 Originally Posted by charybdis You get a few extra crossover points by going from mod 3 to mod 6. That's it. Number theorists are not particularly concerned with the exact positions of the crossover points - especially once the first one has been found - but about the long-term trends. Changing the difference by 1 does not affect these. The difference is usually much larger than 1.
This is my humble task, to find the first crossover point for the mod 6 prime number race. Let the number theorists ponder about the long-term trends after that indeed.

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