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.
Jan Feliksiak graduated from the Monash University Australia, with a degree in Mathematics. From 2002, he has been involved in research in the areas of analytical number theory and differential equations.
Feliksiaks first nonfiction publication tackles one of the great unsolved problems of mathematics.
Riemanns hypothesis, which, among other things, posits the distribution of all prime numbers, represents such a complex problem that even modern mathematical computing software can only perform the calculations to a limited point. As a result, Feliksiaks work is, of necessity, intended for those with a solid understanding of advanced mathematics. However, the author eschews a mathematicians dry tone and instead opts for conversational, direct language, even slipping the occasional colloquialism into his prose-such as an explanation of why an estimation errordoesnt "blow up." The language is technically perfect, to lessen the risk of inaccurately communicating the mathematical steps involved in proving the "very complex problem at hand." At every step, he explains cogently, if briefly, how each theorem represents another small advance toward a better estimate of the prime counting function and, ultimately, an elementary proof of one of the great unsolved problems of mathematics. However, readers without a solid background in number theory will, unfortunately, be lost. As it is, Feliksiaks proposed two-page proof requires more than 100 pages of buildup, even in the compressed language of mathematics. But the author has, at least, managed to make his advanced computations and reasoning accessible to anyone who understands the mathematical terms he employs.
Along the way, he offers tidbits of history about the development of number theory and prior attempts to prove Riemanns hypothesis, establishing a context of other mathematicians past contributions. For readers with a grounding in number theory, the interlocking parts of this mathematical proof come together to create a surprisingly harmonious whole.
- Kirkus Reviews