Application of Data Mining and Mathematical Techniques to the Zeta
Function and the Riemann Hypothesis
Isa Muqattash
Advisor: Dr. Yahdi
2003
The Riemann Zeta function is the analytic continuation of the Zeta function that
Euler introduced throughout his study of the distribution of the prime numbers.
The Riemann Zeta function has two types of zeros: those that are trivial, which
are the even negative integers, and those non-trivial which are complex non-real
and all have real part between zero and one. Riemann conjectured that all the
non-trivial zeros of Zeta have a real part equal to one half; i.e. they lie on the
critical line in the complex plane. This is known as the Riemann Hypothesis (RH);
an open question for about 160 years now, and is considered one of only seven
problems of the millennium. The first 250 billion zeros have been calculated
without any counterexample to RH. Due to the immense support of RH, many
mathematicians take it for granted in order to set new theories in various fields of
mathematics. A proof of RH will validate these theories, in particular about the
distribution of the primes.
To simplify the study of the Riemann Zeta function, we study the embedded
patterns and relations amongst the complex parts of its non-trivial zeros. Using
data mining techniques, we construct an infinite family of recursively-defined
fitting curves for the complex parts of the zeros on the critical line. We then apply
techniques from calculus and analysis to show that those fitting curves are all
asymptotic approximations, and to study their distribution and accuracy.
The methodology used is based on an iterative process of interaction between
the powers of mathematical analysis and data mining techniques. This results in
higher performances and better knowledge about the given data than just using
only one of the two approaches. Such techniques may be applied to various
fields where random data or apparently chaotic phenomena can arise from
recursive or functional processes of interactions or productions.
Moving on to pure mathematics, we introduce the idea of reducing the study from
the entire complex plane, to a study of a real line; achieved by fixing the complex
part of the non-trivial zeros of Riemann’s Zeta function to work within a horizontal
line. Several sufficient conditions are introduced for RH to hold.