20210922, 00:29  #45 
"Καλός"
May 2018
2^{2}×5×17 Posts 
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. Last fiddled with by Dobri on 20210922 at 00:58 
20210922, 06:07  #46 
"Καλός"
May 2018
340_{10} Posts 
Lemma 1: The primecount distance π(x_{2})  π(x_{1}) between two primes x_{1} and x_{2}, x_{1} < x_{2}, for which Δπ(x_{1}) = Δπ(x_{2}) = 0 (two consecutive zeros without other zeros in between) in a binary prime number race of two prime types is an even number, π(x_{2})  π(x_{1}) = 2k.
Proof: The number of primecount steps in one direction must be equal to the number of primecount steps in the opposite direction of the binary prime number race so that starting from Δπ(x_{1}) = 0 the staircase approximation of distinct primecount steps has to end up in Δπ(x_{2}) = 0 for which an even number of steps is required. Lemma 2: The absolute value of an extremum Δπ(x_{m}), x_{1} < x_{m} < x_{2}, between two primes x_{1} and x_{2}, x_{1} < x_{2}, for which Δπ(x_{1}) = Δπ(x_{2}) = 0 (two consecutive zeros without other zeros in between) in a binary prime number race is less than or equal to (π(x_{2})  π(x_{1}))/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). 
20210922, 09:21  #47 
"Καλός"
May 2018
2^{2}×5×17 Posts 
Whoever modified the title of this thread, how do you know that the guess is incorrect?

20210922, 09:51  #48  
"Καλός"
May 2018
2^{2}·5·17 Posts 
Quote:
You abuse your power systematically. Attempting to intimidate and belittle ordinary users, you just reveal your true self. 

20210922, 11:41  #49  
Feb 2017
Nowhere
2^{2}×3×491 Posts 
Quote:
Quote:
3. 10 points for each such statement that is adhered to despite careful correction. Quote:
Quote:
That's 45 points on The PrimePages' Crackpot index right there. 

20210922, 15:09  #50  
"Καλός"
May 2018
2^{2}·5·17 Posts 
Quote:
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. 

20210922, 15:35  #51  
Feb 2017
Nowhere
2^{2}×3×491 Posts 
Quote:
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 1month timeout. 

20220514, 14:34  #52 
"Καλός"
May 2018
2^{2}·5·17 Posts 
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). 
20220514, 16:06  #53 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{2}×3^{4}×11 Posts 
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 
20220518, 14:56  #54 
"Καλός"
May 2018
524_{8} Posts 
In the attached image, the lowest smooth orange curve is the plot of x^{1/2}ln(ln(ln(x)))/(2*ln(x)).
The gray smooth curve above it is the plot of Kx^{1/2}ln(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. 
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