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Keywords: Pi()--Odd Prime Counting Function, Li()--Logrithmatic Integral, ln()--natural log function, and pn and pn+1 are nth prime and (n+1)th prime, respectively, and we count 1 as prime only in the additive sense of number theory.
Here are some important facts we might considered provided Riemann Hypothesis (RH) is true (RH is true!):
*FACT I: |Pi(x) - Li(x)| < ( sqrt(x) * ln(x) ) / (8*[tex]\pi[/tex]) for all x ≥ 2657.
(*See Reference: Schoenfeld, Lowell (1976), "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II",
Mathematics of Computation 30 (134): 337–360, doi:10.2307/2005976, JSTOR 2005976, MR 0457374.)
FACT II: (?) or could it be the Average Prime Gap Error (APGE) associated with the true average gap for all pn and pn+1 in the interval,
[1, x]?
FACT III: What could it be?
Uncertainty Principle (UCP) for the distribution of odd prime numbers:
UCP is |Pi(x) - Li(x)| * APGE ≥ ? for all x ≥ 2657.
Right? or Not even wrong!... !?
A Solution:
Uncertainty Principle (UCP) is |Pi(x) - Li(x)| * |a'(x) - a(x)|.
Calculation:
We let a'(x) = x/Pi(x) where a'(x) is the true average prime gap for all pn and pn+1 in [1, x].
And we let a(x) = x/Li(x) where a(x) is the approximate average prime gap for all pn and pn+1 in [1, x].
|Pi(x) - Li(x)| * |a'(x) - a(x) | < ? where
|Pi(x) - Li(x)| < sqrt(x) * ln(x)/(8*[tex]\pi[/tex]) for all x ≥ 2657, and APGE is |a'(x) - a(x)|, and [tex]\pi[/tex] = 3.14...
|Pi(x) - Li(x)| < sqrt(x) * ln(x)/(8*[tex]\pi[/tex]) implies Pi(x) = Li(x) + r * sqrt(x) * ln(x) / (8 *pi) where -1 < r < 1.
This last equation implies Pi(x) = (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x) )/(8 *[tex]\pi[/tex]).
Therefore, a'(x) = x / Pi(x) = 8 * pi * x / (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x) ).
So, |a'(x) - a(x)| = |8 * [tex]\pi[/tex] * x / (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x)) - x / Li(x) |
= x / Li(x) * | (8 * [tex]\pi[/tex] * Li(x) - 8 * [tex]\pi[/tex] * Li(x) - r * sqrt(x) * ln(x))/ (8 *[tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x)) |
= x / Li(x) * | ( - r * sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) + r * sqrt(x) * ln(x)) |
< x / Li(x) * | ( sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)) |
= x / Li(x) * ( sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)) if r = -1.
Therefore, APGE is |a'(x) - a(x)| < x / Li(x) * | ( sqrt(x) * ln(x))/ (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)) |.
Hence,
|Pi(x) - Li(x)| * |a'(x) - a(x) | < (sqrt(x) * ln(x)) /(8*[tex]\pi[/tex])) * x / Li(x) * ( sqrt(x) * ln(x))/ (8 *[tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x))
= 1/(8 * [tex]\pi[/tex]) * 1/Li(x) * (x * ln(x))^2 / (8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x)).
Thus, we have the following calculation of UCP**:
UCP is |Pi(x) - Li(x)| * |a'(x) - a(x) | < (x * ln(x) )^2 /[8 * [tex]\pi[/tex] * Li(x) * ( 8 * [tex]\pi[/tex] * Li(x) - sqrt(x) * ln(x) ) ].
**Note: The Riemann Prime-Counting Function R(x) will improve UCP substantially because R(x) = Pi(x) since it incorporates all the
non-trivial zeta zeros of the Riemann zeta function in its calculation.
David Cole
(aka primework123)
Please support my research work at: https://www.gofundme.com/david_cole
Thank you! Thank Lord GOD!
http://biblia.com/verseoftheday/image/Ro8.28