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Riemann's R-Function and the Distribution of Primes

The author earlier explored the distribution of primes, where the ratio of the number of primes in the successive intervals defined by the Fibonacci numbers was found to alternate high, low, high, low, etc. from the interval [55, 89] to the interval [121 393, 196 418] [1]. Here Riemann’s R-function is shown to alternately underand over-estimate the number of primes from the interval [55, 89] to the interval [317 811, 514 229]. Below, the values on the right-hand-side reveal whether Riemann’s R-function either underor over-estimates the number of primes in an interval defined by consecutive Fibonacci numbers, where F0 = 0, F1 = 1, F2 = 1, F3 = 2, . . . etc., and where R(x) equals RiemannR[x] in Mathematica. The function π(x) gives the number of primes less than or equal to x.