Incorrect guess based on limited data: Number of Primes 6k-1 > Number of Primes 6k+1

[Dobri]

Guess based on limited data: The number of primes of type 6k-1 is greater than the number of primes of type 6k+1.

Range, π(Range), Number of Primes 6k-1, Number of Primes 6k+1, Difference
1,000,000,000, 50,847,534, 25,424,819, 25,422,713, 25,424,819 - 25,422,713 = 2,106
2,000,000,000, 98,222,287, 49,112,582, 49,109,703, 49,112,582 - 49,109,703 = 2,879
3,000,000,000, 144,449,537, 72,226,055, 72,223,480, 72,226,055 - 72,223,480 = 2,575
...

Note: Primes 2 and 3 are counted in π(Range).

There is a consistent slight difference of a few thousand primes.
Is there a simple way to explain said difference?
Also, please share if there are existing references related to this matter.

(* Wolfram code *)
nrange = 3000000000; nmax = PrimePi[nrange]; tpc = 0; tp5 = 0; tp1 = 0; tpother = 0;
n = 1; While[(n <= nmax), p = Prime[n]; tpc++; If[Mod[p, 6] == 5, tp5++;, If[Mod[p, 6] == 1, tp1++;, tpother++;];]; n++];
Print[tpc, ", ", tp5, ", ", tp1, ", ", tpother];

[charybdis]

You've rediscovered a very well-known phenomenon called Chebyshev's bias. Assuming some strong versions of the Riemann hypothesis, it is known that for most N, there are more primes of the form 2 mod 3 than 1 mod 3 up to N. 1 mod 3 does sometimes take the lead; this first happens at N = 608981813029. The effect relates to the fact that numbers of the form 1 mod 3 can be squares while those of the form 2 mod 3 cannot. In some sense, it's actually the numbers of *primes and prime powers* (edit: with a suitable scaling on the prime powers) that we should expect to be balanced between the two classes, but it's not easy to explain why. See here for an overview of the field.

The ratio (primes 1 mod 3)/(primes 2 mod 3) tends to 1, by de la Vallee Poussin's theorem that for any k the primes are evenly distributed among the residue classes mod k that are coprime to k.

[kriesel]

Occasionally, 6k+1 will be exposed to possible factoring by one higher prime than 6k-1 is, as least factor > 1. For example 167 is prime and = 12.9228...; 169=13².

[Dr Sardonicus]

In the dim past, I posted here, and recommended reading this paper, which is the same as the one linked to in the preceding post to this thread, though at a different URL.

[Dobri]

Originally Posted by charybdis

You've rediscovered a very well-known phenomenon called Chebyshev's bias. Assuming some strong versions of the Riemann hypothesis, it is known that for most N, there are more primes of the form 2 mod 3 than 1 mod 3 up to N. 1 mod 3 does sometimes take the lead; this first happens at N = 608981813029. The effect relates to the fact that numbers of the form 1 mod 3 can be squares while those of the form 2 mod 3 cannot. In some sense, it's actually the numbers of *primes and prime powers* (edit: with a suitable scaling on the prime powers) that we should expect to be balanced between the two classes, but it's not easy to explain why. See here for an overview of the field.

The ratio (primes 1 mod 3)/(primes 2 mod 3) tends to 1, by de la Vallee Poussin's theorem that for any k the primes are evenly distributed among the residue classes mod k that are coprime to k.

Thanks, charybdis, this was useful. It would take several days to reach 608981813029 on a Raspberry Pi 4B device with a compiled Wolfram script. Out of curiosity, I may continue further for some time to observe the tendency after the reversal point.

[LaurV]

The material is nicely presented, however I tend to be careful when I see Granvilles' name (I don't like him much, after many gaffes he did, and after the story with Beal).

Yeah, the two number lines cross infinitely times, however, one "team" is leading "almost" the whole time. Which reminds me of the old joke with two old guys talking on a bench in the park, "how's your sex life these days", "oh, no problem I have sex almost every day", "I don't believe you, I only have it two or three times per year, you are not serious", "of course I am serious, I had sex almost every day", "how come?", "well, Monday I almost had sex, Tuesday I almost had sex, Wednesday I almost had sex, Thursday..."

[RomanM]

100+ years old talk to Doctor

-friend of mine tell me, that he can have sex in row at the one night, and I can not, Help me!
-Ok, its very easy to Help! You just can tell him that You can do the same too!

[Dobri]

The original paper by Bays and Hudson (1978, available in JSTOR, see https://www.jstor.org/stable/2006165...65734d170a779e) has no mention of several cases for small primes when the numbers of primes of the two types π_3,2(x) and π_3,1(x) are equal.

Actually, π_3,2(x) = π_3,1(x) for x = 2, 3, 7, 13, 19, 37, 43, 79, 163, 223 and 229:
π_3,2(2) = π_3,1(2) = 0 (trivial case)
π_3,2(3) = π_3,1(3) = 0 (trivial case)
π_3,2(7) = π_3,1(7) = 1
π_3,2(13) = π_3,1(13) = 2
π_3,2(19) = π_3,1(19) = 3
π_3,2(37) = π_3,1(37) = 5
π_3,2(43) = π_3,1(43) = 6
π_3,2(79) = π_3,1(79) = 10
π_3,2(163) = π_3,1(163) = 18
π_3,2(223) = π_3,1(223) = 23
π_3,2(229) = π_3,1(229) = 24

It seems that the problem is still open for x -> Infinity as there is no strict proof.

[Dobri]

Originally Posted by LaurV

Yeah, the two number lines cross infinitely times, however, one "team" is leading "almost" the whole time.

Indeed, it is unclear (at least to me) what happens at x -> Infinity.
It could be π_3,2(x) = π_3,1(x) + C for some small non-zero constant C. Or C=0?

[Dobri]

Originally Posted by Dr Sardonicus

In the dim past, I posted here, and recommended reading this paper, which is the same as the one linked to in the preceding post to this thread, though at a different URL.

Originally Posted by Dr Sardonicus

According to Prime Races [which I recommend reading], 1 mod 6 doesn't take the lead until p = 608,981,813,029.

The limit of the ratio π_3,2(x) / π_3,1(x) would tend to 1 as x -> Infinity for the infinite number of reversals.
What I am seeking an answer to is if π_3,2(x) = π_3,1(x) + C for some small non-zero constant C.

[charybdis]

Originally Posted by Dobri

π_3,2(2) = π_3,1(2) = 0 (trivial case)
π_3,2(3) = π_3,1(3) = 0 (trivial case)
π_3,2(7) = π_3,1(7) = 1
...

Don't forget 2 is a prime of the form 2 mod 3.

Originally Posted by Dobri

Indeed, it is unclear (at least to me) what happens at x -> Infinity.
It could be π_3,2(x) = π_3,1(x) + C for some small non-zero constant C. Or C=0?

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.