Holy Grail of Mathematics Pursued in 'The Symphony of Primes, Distribution of Primes and Riemann's Hypothesis'
The Riemann's Hypothesis - It has become the Holy Grail of Mathematics. Of great mystery and controversy, it has been mentioned in Ron Howard's film "A Beautiful Mind," in a Season 1 episode of the television crime drama "NUMB3RS," and a novel, titled "Life After Genius." Mathematician Jan Feliksiak labors to bring the focus back to the more serious, disciplined and scholarly attempt to contribute to this quest with his newly published book, titled The Symphony of Primes, Distribution of Primes and Riemann's Hypothesis.
"In November 1859 Bernhard Riemann presented his memoir "On the number of primes under a given magnitude" in which he made a conjecture regarding the ordered behaviour of zeroes of the Riemann's zeta function. This conjecture later became known as the Riemann's Hypothesis," Feliksiak writes. "The quest to prove R-H began and with time it became the Holy Grail of Mathematics."
This book presents research results concerning the distribution of prime numbers. The first major result discussed is the supremum for the maximal prime gaps. By an implementation of a binomial coefficient the maximal prime gaps supremum bound is proved, simultaneously establishing the infimum for primes in the short interval.
Subsequently, a novel application of the theory of the primorial function establishes the tailored logarithmic integral, which is a superior adaptation of the classical Gauss' logarithmic integral. The tailored integral due to its radically improved accuracy over the Gauss' logarithmic integral, constitutes the supremum bound of estimation of the prime counting function. It presents the possibility to estimate the prime counting function with unprecedented accuracy.