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Old 2021-09-22, 06:07   #46
Dobri
 
"刀-比-日"
May 2018

5·47 Posts
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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(xm), x1 < xm < 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).
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