Problems & Puzzles: Conjectures

Conjecture 82. �Average of log Dn / log(logPn) equal R = 0,877 08...

Alain Rochelle sent the following conjecture, on March 17, 2019:

Denoting D(Pm) = P(n) - P(m) the difference between two consecutive primes Pm and Pn (with n=m+1) and denoting logX the natural logarithm of the number X, we consider the set of N consecutive primes included between X and X+X/(logX)^2.

The mathematic notation is :

X < P1�< P2�< ...... < PN-1�< PN�< X+X/(logX)^2 < PN+1

For X to infinite, denoting Dn = Pn+1�- Pn, we conjecture that the average of log Dn / log(logPn) equal R = 0,877 08...�

Denoting SIGMA[...] as the sum for n=1 to n=N, we write :

A(X) = (1/N) x SIGMA[ logDn / log(logPn) ] ---> R� (for X to infinite)

In practice, we verify by computer computation for increasingly values of X :MAG[A(X) - R] is of the order of log(X) / SR(X) �� (with MAG[ ] the magnitude of [ ] and SR(X) the square root of X)

For example, I compute :

X = 2 x 10^8 ; N = 28 693 ; A(X) = 0,877 20

X = 4 x 10^8 ; N = 51 389 ; A(X) = 0,876 91

X = 6 x 10^8 ; N = 72 791 ; A(X) = 0,876 78

X = 8 x 10^8 ; N = 93 010 ; A(X) = 0,876 69

X = 9,8 x 10^8 ; N = 110 340 ; A(X) = 0,877 51 and logX/SR(X) = 0,000 66

The purpose consists to verify if this conjecture is true for large values of X :

X = 10^9 ; X = 2 x 10^9 ; X = 10^10 ; X = 10^11 ; X = 10^12� and so on ...

Q. Send your own verifications

Alain Rochelli wrote on Oct 12, 2019

For Conjecture 82, I computed the first fifty million primes.

The result is :

P(1) = 2 ; P(50 000 000) = 982 451 653

Average of log Dn / log(logPn) equal 0,877 095

Who is the Puzzler to compute the next fifty million primes ?

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On June 7, 2021, Alain wrote again:

Using the same mathematical model as for Conjecture 89 (Cramer 's model with the independence of gaps between consecutive primes), we get the value of� R equal to 1 for P(n) to infinite, with P(n) or Pn the nth prime and Dn or Dp(n) = p(n+1) - p(n) the corresponding difference.
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More precisely, we have (with AV the average value) the mathematical formula :
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AV(Rn) = AV[ logDn / log(logPn) ] = 1 - (G / log(logPn)) where G is Euler 's constant, equal to 0,577215665...
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If we consider Dp as the random variable representing the difference between consecutive primes, the mathematical wording is :
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Around Pn, Rn is equal to the average exponent of logPn to get Dn ( i.e. : Dn = logPn ^ Rn )
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Also AV(Rn) = AV[ log(Dn) / log(logPn) ] = AV{ 1 + [(logDn-log(logPn)) / log(logPn)] } = 1 + {AV[log(Dn/logPn)] / log(logPn)} for Pn to infinite, because log(logPn) is practically a constant.
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Now using the main conjecture saying that AV[log(Dn/logPn)] is converging to - G for Pn to infinite, we then obtain AV(Rn) = 1-[G/log(logPn)] as stated previously.
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This is equivalent to AV( logDn ) = log(logPn) - G because logDn = Rn * log(logPn)
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It follows that the geometric mean GM of Dp is equal to logp * e^(-G) where e^(-G) = 1/e^G = 1 / 1,781072418
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noted again by the wonderful formulation GM( Dp ) = logp / e^(G) = AV( Dp ) / 1,78107...
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For mathematicians, it 's possible that the proof of this conjecture is of the same level of difficulty as for the Riemann 's hypothesis.
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By computer calculation, we need to take into account local variations and go towards very large primes to obtain meaningful results.
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For example, it 's necessary to consider primes around 10 ^ 500 to reach R equal to 0,92.

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