# Nigerian Scholar solves 156-Year-old problem in Maths

By Rotimi Ojomoyela

Dr. Opeyemi Enoch

Ado-Ekiti, Nigeria (Vanguard) — The 156 years old Riemann Hypothesis, the most important problem in Mathematics has been successfully solved by Nigeria Scholar, Dr Opeyemi Enoch.

With this breakthrough, Dr Enoch, who teaches at the Federal University, Oye Ekiti (FUOYE), has become the fourth egghead to resolve one of the seven Millennium Problems in Mathematics.

The Kogi State-born mathematician had, before now, worked on mathematical models and structures for generating electricity from sound, thunder and Oceanic bodies.

A statement in Ado Ekiti yesterday said Dr Enoch presentation of the Proof on November 11, 2015 during the International Conference on Mathematics and Computer Science in Vienna, Austria becomes more symbolic coming on the exact day and month 156 years after the problem was delivered by a German Mathematician in 1859.

The Riemann Zeta Hypothesis is one of the seven Millennium problems set forth by the Clay Mathematics Institute with a million Dollar reward for each solved problem for the past 16 years.

According to the statement, “Dr Enoch first investigated and then established the claims of Riemann. He went on to Consider and to correct the misconceptions that were communicated by Mathematicians in the past generations, thus paving way for his solutions and proofs to be established.

“He also showed how other problems of this kind can be formulated and obtained the matrix that Hilbert and Poly predicted will give these undiscovered solutions. He revealed how these solutions are applicable in cryptography, quantum information science and in quantum computers,” it stated.

Three of the problems had been solved and the prizes given to the winners. This makes it the fourth to be solved of all the seven problems.

Dr Enoch had previously designed a Prototype of a silo for peasant farmers and also discovered a scientific technique for detecting and tracking someone on an evil mission.

Enoch has succeeded in inventing methods by which oil pipelines can be protected from vandalism and he is currently working on Mathematical approaches to Climate Change.

# Riemann hypothesis

In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the number of primes π(x) less than a given number x. His formula was given in terms of the related function

\begin{align}\Pi(x) &= \pi(x) +\tfrac{1}{2}\pi(x^{\frac{1}{2}}) +\tfrac{1}{3}\pi(x^{\frac{1}{3}}) +\tfrac{1}{4}\pi(x^{\frac{1}{4}}) \\ &\ \ \ \ +\tfrac{1}{5}\pi(x^{\frac{1}{5}}) +\tfrac{1}{6}\pi(x^{\frac{1}{6}}) +\cdots\end{align}

which counts the primes and prime powers up to x, counting a prime power pn as 1/n of a prime. The number of primes can be recovered from this function by

\begin{align} \pi(x) &= \sum_{n=1}^{\infty}\frac{\mu(n)}{n}\Pi(x^{\frac{1}{n}}) \\ &= \Pi(x) -\frac{1}{2}\Pi(x^{\frac{1}{2}}) -\frac{1}{3}\Pi(x^{\frac{1}{3}}) -\frac{1}{5}\Pi(x^{\frac{1}{5}}) \\ &\ \ \ \ +\frac{1}{6}\Pi(x^{\frac{1}{6}}) -\cdots, \end{align}

where μ is the Möbius function. Riemann’s formula is then

\begin{align}\Pi_0(x) &= \operatorname{Li}(x) - \sum_\rho \operatorname{Li}(x^\rho) -\log(2) \\ &\ \ \ \ +\int_x^\infty\frac{dt}{t(t^2-1)\log(t)}\end{align}

where the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits:

$\Pi_0(x) = \lim_{\varepsilon \to 0}\frac{\Pi(x-\varepsilon)+\Pi(x+\varepsilon)}2.$

The summation in Riemann’s formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function Li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral

$\operatorname{Li}(x) = \int_0^x\frac{dt}{\log(t)}.$

The terms Li(xρ) involving the zeros of the zeta function need some care in their definition as Li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ ln x). The other terms also correspond to zeros: the dominant term Li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see Riesel & Göhl (1970) or Zagier (1977).

This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their “expected” positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.