[QUOTE=charybdis;586577]Don't forget 2 is a prime of the form 2 mod 3.[/QUOTE]
My bad, the Wolfram script is actually testing the [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) and [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) cases.

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

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

[QUOTE=bhelmes;586579]Perhaps this is helpful for you:

p 1 3 5 mod 6
...[/QUOTE]
Thanks, this is of limited use as the task is to study the exact locations of the reversal points for which the sign of [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) flips to the opposite, or eventually [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) = 0 with or without a subsequent sign reversal for larger primes. There is no need of considering the only prime [I]π[/I][SUB]6,3[/SUB]([I]x[/I]) = 3 for said task.

[QUOTE=charybdis;586577]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.[/QUOTE]
Indeed, the question is what is happening [B]on the average[/B] (rather than at distinct sampling points or limited sampling intervals) when the number of reversal points approaches infinity.

It should be noted that the prime number race [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) is not the same with the race [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) and their respective reversal points might differ.
Therefore, unless a reference is found to prove that the reversal points of the race [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) 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.

[QUOTE=Dobri;586593]Indeed, the question is what is happening [B]on the average[/B] (rather than at distinct sampling points or limited sampling intervals) when the number of reversal points approaches infinity.[/QUOTE]

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=Dobri;586594]Therefore, unless a reference is found to prove that the reversal points of the race [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) have been studied in the past, the said race is a new topic.[/QUOTE]

This has been investigated. See OEIS [URL="https://oeis.org/A096449"]A096449[/URL].

[QUOTE=charybdis;586595]This has been investigated. See OEIS [URL="https://oeis.org/A096449"]A096449[/URL].[/QUOTE]
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.

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.

[QUOTE=charybdis;586597]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.[/QUOTE]
There appears to be a significant difference. For instance, the list of [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) = [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) 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.

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.

[QUOTE=charybdis;586600]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.[/QUOTE]
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.