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

[Dobri]

Please share if knowing about a reference to a previous study investigating computationally the exact local maxima and minima of the Δπ differences of prime number races.
I am aware about a few papers and several OEIS sequences concerned with Δπ = -1, 0, and 1 differences only.

[Dobri]

Lemma 1: The prime-count distance π(x₂) - π(x₁) between two primes x₁ and x₂, x₁ < x₂, for which Δπ(x₁) = Δπ(x₂) = 0 (two consecutive zeros without other zeros in between) in a binary prime number race of two prime types is an even number, π(x₂) - π(x₁) = 2k.
Proof: The number of prime-count steps in one direction must be equal to the number of prime-count steps in the opposite direction of the binary prime number race so that starting from Δπ(x₁) = 0 the staircase approximation of distinct prime-count steps has to end up in Δπ(x₂) = 0 for which an even number of steps is required.

Lemma 2: The absolute value of an extremum Δπ(x_m), x₁ < x_m < x₂, between two primes x₁ and x₂, x₁ < x₂, for which Δπ(x₁) = Δπ(x₂) = 0 (two consecutive zeros without other zeros in between) in a binary prime number race is less than or equal to (π(x₂) - π(x₁))/2.
Proof: The absolute value of the prime count of the extremum cannot exceed k in Lemma 1.
In the ideal case with only one extremum, the absolute value of said extremum (minimum or maximum) = k.

Therefore, the knowledge of the distribution of consecutive zeros Δπ(x) = 0 can be used to estimate in first approximation the Max[Δπ(x)] distribution and plot an ideal bar graph (assuming a single extremum between consecutive zero Δπ(x) points).

The Littlewood theorem states that Δπ(x) can take arbitrarily large values toward infinity but does not explicitly specify how frequently this could happen in comparison with other Δπ(x) values. Therefore, analyzing the prime number race results from the past, and also performing recounts to obtain the exact local minima and maxima distributions, does matter.

An ideal bar graph (assuming a single extremum between two consecutive zero points) on the basis of the 85,508 zero Δπ(x) points in A096629 can be plotted to show in first approximation the shape of the distribution of the extrema of Δπ(x).

[Dobri]

Whoever modified the title of this thread, how do you know that the guess is incorrect?

[Dobri]

Originally Posted by Batalov

P.S. Your "blogging" belongs in Blogorrhea section (where the blogger is the only one reading). Blog as much as you want.

You abuse your power systematically. Attempting to intimidate and belittle ordinary users, you just reveal your true self.

[Dr Sardonicus]

Originally Posted by Dobri

The number of primes of type 6k-1 is greater than the number of primes of type 6k+1.

Originally Posted by Dobri

Whoever modified the title of this thread, how do you know that the guess is incorrect?

2. 5 points for every statement that is clearly vacuous, logically inconsistent, or widely agreed on to be false.
3. 10 points for each such statement that is adhered to despite careful correction.

Originally Posted by Dobri

The Littlewood theorem states that Δπ(x) can take arbitrarily large values toward infinity but does not explicitly specify how frequently this could happen in comparison with other Δπ(x) values. Therefore, analyzing the prime number race results from the past, and also performing recounts to obtain the exact local minima and maxima distributions, does matter.

7. 10 points for stating that your ideas are of great financial, theoretical and/or spiritual value.

Originally Posted by Dobri

You abuse your power systematically. Attempting to intimidate and belittle ordinary users, you just reveal your true self.

26. 20 points for each complaint that the list moderators are out to get you or are blocking your substantial and useful posts.

That's 45 points on The PrimePages' Crackpot index right there.

[Dobri]

Originally Posted by Dr Sardonicus

2. 5 points for every statement that is clearly vacuous, logically inconsistent, or widely agreed on to be false.
3. 10 points for each such statement that is adhered to despite careful correction.
7. 10 points for stating that your ideas are of great financial, theoretical and/or spiritual value.
26. 20 points for each complaint that the list moderators are out to get you or are blocking your substantial and useful posts.
That's 45 points on The PrimePages' Crackpot index right there.

Do you imply that the the initial guess is false, and you arrived at this conclusion after a careful consideration?

Then you must be able to provide tangible evidence, preferably works published by others, to support your careful consideration, and explain yourself clearly without speaking in riddles or giving 'homework'.

Please do not put words in my mouth as it was never stated in this thread that the "ideas are of great financial, theoretical and/or spiritual value."

If you are wrong about points #2, #3, and #7, then adding point #26 does not make sense.

[Dr Sardonicus]

Originally Posted by Dobri

Do you imply that the the initial guess is false, and you arrived at this conclusion after a careful consideration?
<snip>

Objection! Already asked and answered!

You have already publicly acknowledged references proving your guess is wrong that were offered, e.g. in this post and this post to this thread.

Objection sustained. I find you guilty of trolling, and will see about imposing a 1-month timeout.

[Dobri]

This update contains images of the values of the absolute minima and maxima (up to a prime x on the horizontal axis) of the prime race ∆(x) = 𝜋_6,5(x) - 𝜋_6,1(x) for the first 185,000,000,000 primes.
The ongoing exhaustive computation is getting closer to the next equilibrium point ∆(x) = 0 listed in OEIS A096449 (see post #38, https://mersenneforum.org/showpost.p...0&postcount=38).

[sweety439]

OEIS sequences references of your problem:

https://oeis.org/A007350
https://oeis.org/A007352
https://oeis.org/A199547
https://oeis.org/A306891
https://oeis.org/A038698
https://oeis.org/A112632
https://oeis.org/A066520
https://oeis.org/A321856
https://oeis.org/A275939
https://oeis.org/A326615
https://oeis.org/A306499
https://oeis.org/A306500

[Dobri]

In the attached image, the lowest smooth orange curve is the plot of x^1/2ln(ln(ln(x)))/(2*ln(x)).
The gray smooth curve above it is the plot of Kx^1/2ln(ln(ln(x)))/(2*ln(x)) with an empirically chosen coefficient K = 28.
Correction to my previous post #52, https://mersenneforum.org/showpost.p...6&postcount=52: The OEIS sequence is A096629.