MATHS BITE: Apéry’s Constant

Apéry’s constant is defined as the number

{\displaystyle {\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)\end{aligned}}}

where ζ is the Riemann Zeta Function.

This constant is named after the French mathematician Roger Apéry who proved that it was irrational in 1978. However it is still unknown whether or not it is transcendental.


The Basel Problem asked about the convergence of the following sum:
Screen Shot 2017-06-10 at 2.16.05 PM.png

In the 18th century, Leonhard Euler proved that in fact it did – to π^2/6. However, the limit of the following sum remained unknown:Screen Shot 2017-06-10 at 2.19.28 PM.png

Although mathematicians made some progress, including Euler who calculated the first 16 decimal digits of the sum, it was not known whether the number was rational or irrational, until Apéry.

Furthermore, it is currently not known specifically whether any other particular ζ(n), for n odd, is irrational. “The best we’ve got is from Wadim Zudilin, in 2001, who showed that at least one of ζ(5), ζ(7), ζ(9), ζ(11) must be irrational, and Tanguy Rivoal, in 2000, who showed that infinitely many of the ζ(2k+1) must be irrational.”

M x


Physics and Riemann Hypothesis?

Researchers have recently discovered that solutions to the Riemann zeta function correspond to the solutions of another function that may make it easier to solve the Riemann hypothesis.

Dorje Brody, a mathematical physicist at Brunel University London, says that “to our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues [‘solutions’ in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function“.

Now what remains to be proven is that all of the eigenvalues are real numbers rather than imaginary ones.

This newly discovered operator has close ties with quantum physics. In 1999, Michael Berry and Jonathan Keating made the conjecture (now called the Berry-Keating conjecture) that if such an operator exists, then it should correspond to a theoretical quantum system with particular properties. However, no one has ever found such a system before now.

In general, mathematicians are optimistic that the eigenvalues are actually real, and there is a strong argument for this based on PT symmetry (a concept from quantum physics which says that you can change the signs of all four components of space-time and the result will look the same as the original).


Click here for more on the Riemann hypothesis. M x

Math’s Bite: Mills’ Constant

In number theory, Mills’ theorem states that there exists a real constant A such that |_A^(3^n)_| is prime for all positive integers n (note that this is a floor function). Mills’ constant is defined as the smallest real positive number such that Mills’ theorem is true.

This constant is named after William H. Mills, who proved the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps in 1947.

If Riemann’s hypothesis is true, Mills’ constant is approximately:


Mills’ Primes

The primes generated using Mills’ constant are known as Mills’ primes:

“If ai denotes the ith prime in this sequence, then ai can be calculated as the smallest prime number larger than a_{i-1}^3. In order to ensure that rounding A^{3^n} produces this sequence of primes, it must be the case that a_i < (a_{i-1}+1)^3.”

In 2005, Caldwell and Cheng computed 6850 base 10 digits of Mills’ constant under the assumption that the Riemann hypothesis is true.

Hope you enjoyed this installment of ‘Math’s Bite’! M x

The Millennium Prize Problems

As a young girl, I always dreamed of winning a Nobel Prize. Now, you can imagine my disappointment when, as a slightly older teenager interested in maths over science, I discovered the sad truth that in fact there was no Nobel Prize in mathematics. As soon as I learnt of The Millennium Prize Problems (thanks dad!), they quickly became my dream – I was going to prove one when I was older. So inspired by this dream, I have read books and watched countless documentaries on the proof of Fermat’s Last Theorem, Poincaré’s Conjecture and so on.

But what are the Millennium Prize Problems?

The Millennium Prize Problems are seven problems in mathematics, set by the Clay Mathematics Institute at the start of this century – 2000. The Prizes were created to record some of the most challenging problems in mathematics, to increase the awareness of mathematics to the general public and emphasise the importance of working towards a solution of the most difficult problems. Six problems still remain unresolved.

Although for many mathematicians, money is not their motivation for the work that they do, it is worth saying that $1 million is given to the individual who solves each problem.

The Millennium Prize Problems are:

  • Poincaré Conjecture (solved)
  • Birch and Swinnerton-Dyer Conjecture
  • Hodge Conjecture
  • Navier-Stokes Equation
  • P vs NP Problem
  • Riemann Hypothesis
  • Yang-Mills and Mass Gap

Poincaré Conjecture

Formulated in 1904 by Henri Poincaré, the Poincaré conjecture deals with the branch of math called topology. It states that  “every simply connected closed three-manifold is homeomorphic to the three-sphere”. What does this mean?

poincare conjectureImagine stretching a rubber band around the surface of a ball. If we shrink this rubber band, it will shrink down to a point without tearing it or allowing it to leave the surface of the ball. This means that the surface of the ball is ‘simply connected’. A good example of an object that isn’t simply connected is a doughnut, as there is no way of shrinking the rubber band to a point without breaking the band or the doughnut. Poincaré saw that the 2-dimensional sphere is characterised by this property, and so any 2-dimensional object that was simply connected could be deformed and shaped into a sphere, without breaking it apart. He then went on to conjecture that the same would be true in 3-dimensions.

Grigori Perelman
Grigori Perelman

This proved extremely difficult to show; it took almost a century for a solution to be found. In 2002 and 2003, Russian mathematician Grigori Perelman posted a series of papers on arXiv, which have withstood intense scrutiny by the mathematical community for the past four years. In his proof, Perelman used Richard Hamilton’s theory of Ricci flow, and made use of results on spaces of metrics due to Cheeger, Gromov, and Perelman himself.

I would recommend watching this video for more information.

Birch and Swinnerton-Dyer Conjecture

This conjecture is in the field of number theory and is named after Bryan Birch and Peter Swinnerton-Dyer who developed the conjecture during the first half of the 1960s. As of 2015, only special cases of the conjecture have been proven correct.

The conjecture focuses on describing whole number solutions to algebraic equations like:

Screen Shot 2015-09-18 at 4.55.58 PM

Although Euler managed to find the complete solution for this equation, with more complicated equations this becomes very difficult; in 1970 Yu. V. found that there is no general method to finding solutions to these types of equations.

Birch and Swinnerton-Dyer
Birch and Swinnerton-Dyer

When the solutions to these equations are the points of an abelian variety (a complete algebraic variety whose points form a group), the Birch and Swinnerton-Dyer conjecture says that the size of the group of rational points is related to the behaviour of an associated zeta function ζ(s) near the point s=1. Specifically, it asserts that if ζ(1) is equal to 0, then there are an infinite number of rational points or solutions, and conversely, if ζ(1) is not equal to 0, then there is only a finite number of such points.

Hodge Conjecture

William Hodge
William Hodge

Formulated by Scottish mathematician William Hodge between 1930 and 1940, this conjecture questions how well we can approximate the shape of a given object by gluing together simple geometric building blocks of increasing dimension. Due to the usefulness of this technique, it has been generalised in many different ways, thus developing powerful mathematical tools exploited by mathematicians. However, unfortunately the geometric origins became lost in these generalisations.

The Hodge conjecture states that for particularly ‘nice’ spaces, which are called projective algebraic varieties, the pieces called Hodge cycles are actually combinations of geometric pieces called algebraic cycles.


Navier-Stokes Equation

This problem in mathematical physics deals with the motion of fluid and viscous fluids, for example, waves and turbulent air currents. The solutions to the Navier-Stokes equations are believed to explain and predict the motion of such fluids. However, although these equations were written down in the 19th century, our understanding of them is still primitive and hence the challenge in to make significant progress toward a solid mathematical theory.

P vs NP Problem

I’ve always found this problem fascinating, as I genuinely do not know how you would even start to think of a solution it. It is a problem in computer science and asks: can every solved problem whose answer can be checked quickly by a computer also be quickly solved by a computer? So, the problem is to determine whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure.

P and NP refer to thee two types of maths problems:1000px-P_np_np-complete_np-hard.svg

  • P problems are fast for computers to solve,and so are considered “easy”.
  • NP problems are fast and therefore easy for a computer to check, but are not necessarily easy to solve.

So far no one has managed to prove that there is a problem whose answer cannot be feasibly generated with the help of a computer.

Riemann Hypothesis

This is the Millennium Prize problem that I find the most intriguing, as it involves prime numbers. To me, Riemann was a genius.

Although the distribution of such prime numbers among all natural numbers does not follow any regular pattern, Riemann noted that the frequency of prime numbers is very closely related to the behaviour of an elaborate function called the Riemann Zeta function:

Screen Shot 2015-09-18 at 5.38.47 PM

In 1859, Riemann hypothesised that all non-trivial solutions of the equation ζ(s) = 0 lie on a certain vertical straight line: real-part = 1/2. riemann1

This has been checked for the first 10,000,000,000 solutions. A proof that it is true for every interesting solution would shed light on many of the mysteries surrounding the distribution of prime numbers, and could possibly be disastrous for RSA encryption.

Although it is hard to explain and understand, I would recommend reading ‘Music of the Primes’ by Marcus du Sautoy as this book helped me truly grasp this problem. This video is also worth watching.

Yang-Mills and Mass Gap

This is another problem in the field of mathematical physics, which deals with quantum physics. Yang and Mills introduced a way to describe elementary particles using structures that also occur in geometry, which is called Quantum Yang-Mills theory, and is now the foundation to most elementary particle theory. Its predictions have been test by experiment, but the mathematical foundations are still shaky and remain unclear. The success of the theory relies on a property called the ‘mass gap’, which is that quantum particles have positive masses, even though the classical waves travel at the speed of light. The problem is to therefore establish the existence of the Yang-Mills Theory and mass gap theoretically using mathematics.

Don’t be discouraged if you don’t understand fully what these mean, I myself don’t either! I’m no expert, I just personally find them fascinating and I love learning about them.

Let me know which problem you find the most interesting! M x

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