constant

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.

History

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

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Maths Bite: Hubble’s Constant

In today’s post I want to take a small diversion into the realm of Physics, in particular Astrophysics, to look at an extremely significant constant: Hubble’s constant.

Edwin Hubble measured the speed of galaxies and their distance from Earth and obtained the following graph:

As the graph is a straight line through the origin it shows that velocity is directly proportional to the distance from Earth. This is known as ‘Hubble’s law’.

Measuring Distances

Hubble’s Constant can be used to measure astronomical distances, which are too big to be measured by parallax or by using standard candles.

Hubble’s ConstantHubbleLaw.gifhas a value of 2.3 x 10-18 s^-1 or 72 km s^–1 Mpc^–1

[Mpc = megaparsec = 3.26 million light years]

As:

 HubbleLaw.gif

and v = zc (where z is the redshift of the galaxy):

redshift equation5

we can find the distance for any distant galaxy, provided we can measure its redshift (z).

Assumptions

We need to assume the straight line remains linear as the redshift becomes bigger and bigger – Hubble’s law holds universally.

This is not true, for example, with Hooke’s law, as it has a limit of proportionality (elastic limit).

Age of the Universe

All distant objects are moving away from us, suggesting that the Universe as a whole is expanding. If we turn back time, then the Universe would contract to a single point. This moment is called the Big Bang.

If we can find the Hubble constant, it will tell us how quickly the Universe is expanding, and from this we can work out how old our universe is.

If the universe was created at a time T ago, for a galaxy that has been moving away from us at a steady rate v for a time T, its distance d from us will now be vT.

Hubble’s Law tells us v = Hd, so v = HvT, which gives us HT = 1. Hence, the age of the universe can be given by:

T = 1/H

This gives an estimate of 14 billion years to 2 significant figures.

However, there are great uncertainties involved with this estimate:

  • Gravitational forces will mean that the present rate of expansion is less than in the past, so T < 1/H
  • Although the value for Hubble’s constant has become more accurate since the launch of the Hubble Space Telescope, the current value is only considered accurate to within 5%, so there is an uncertainty to the value for T.

Fate of the Universe?

  • CLOSED UNIVERSE: since gravity works against expansion, if the density were large enough then the expansion would stop and the universe would collapse in a ‘big crunch’. (Ω > 1)
  • OPEN UNIVERSE: If the density is small enough, then the expansion would continue forever – steady increase in Hubble’s constant. (Ω < 1)

Hope you enjoyed the post. Let me know what you think of more physics-based posts! M x

Math’s Bite: Euler-Mascheroni constant

The Euler-Mascheroni Constant appears in analysis and number theory, and is denoted by γ. It is defined as the limit of the difference between the harmonic series and the natural logarithm:

\begin{align}
\gamma &= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)\\
&=\int_1^\infty\left(\frac1{\lfloor x\rfloor}-\frac1{x}\right)\,dx.
\end{align}

Note that the ⌊x⌋ represents a floor function.

It has not been shown whether γ is algebraic or transcendental. In fact, it is not even known whether it is irrational. If γ is a simple fraction a/b, it is known that b>10^(242080) (a result given by T. Papanikolaou).

Hilbert said that the irrationality of γ is an unsolved problem that seems “unapproachable” and in front of which mathematicians stand helpless.

To find out more places where this constant arises, click here.

History

This constant was first defined by Euler in a paper published in 1734 entitled ‘De Progressionibus harmonicis observationes’. In the paper, Euler used the letters C or O for the constant. In 1790, Mascheroni used the notations A or a for this constant. It is said that the symbol γ was chosen at a later time due to its relation to the gamma function; γ is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1:

-\gamma = \Gamma'(1) = \Psi(1).

Approximations

The constant has a numerical value of 0.57721566490153286060651209008240243104215933593992 to 50 decimal places.

A beautiful approximation to is γ given by

 pi/(2e)=0.57786367...

which is correct to three decimal places.

An approximation given by P Galliani, involving unit fractions:

 1/2+1/(23)+1/(37)+1/(149)-1/(968625)=0.5772156649012...,

is good to 12 decimal places.

Furthermore, Barbosa gave the pandigital approximation:

 gamma approx -(e^(-6^3/9))/e+(exp(-exp(e^(.8)))+.4)/(ln2)+(ln5)/(10^7)

which is good to 13 decimal places!

Hope you liked today’s post. 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:

 theta=1.306377883863080690...

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

Maths Bite: Conway’s Constant

Look-and-Say Sequences

A Look-and-Say sequence was first introduced and analysed by John Conway. An example of such series is:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211….

To generate the next number in the sequence from the previous term, read off the digits of the previous number, counting the number of digits in groups of the same digit. For example:

  • 1 is read as ‘one 1’ = 11
  • 11 is read as ‘two 1s’ = 21
  • 21 is read as ‘one 2, one 1’ = 1211

If we start with any digit x from 0 to 9, then x will remain the last digit of the sequence. When x does not equal 1, the sequence is as follows:

x, 1x, 111x, 311x, 13211x, 111312211x, 31131122211x…

The Conway sequence, named by Vardi in 1991, is a look-and-say sequence with the starting digit 3.

Growth in Length and Conway’s Constant

The terms of the sequence eventually grow in length about 30% per generation. If Ln denotes the number of digits in the n-th term of the sequence, the limit of the ratio

\frac{L_{n + 1}}{L_n}

is Conway’s constant:

\lim_{n \to \infty}\frac{L_{n+1}}{L_{n}} = \lambda

where λ = 1.303577269034…

 

Source: Wikipedia

Conway’s constant is the unique positive real root of the following polynomial:

\begin{align}
&\,\,\,\,\,\,\,  x^{71}   &&  &&- x^{69}   &&- 2x^{68}  &&- x^{67}   &&+ 2x^{66}  &&+ 2x^{65}  &&+ x^{64}   &&- x^{63} \\
&- x^{62}  &&- x^{61}   &&- x^{60}   &&- x^{59}   &&+ 2x^{58}  &&+ 5x^{57}  &&+ 3x^{56}  &&- 2x^{55}  &&- 10x^{54} \\
&- 3x^{53} &&- 2x^{52}  &&+ 6x^{51}  &&+ 6x^{50}  &&+ x^{49}   &&+ 9x^{48}  &&- 3x^{47}  &&- 7x^{46}  &&- 8x^{45}  \\
&- 8x^{44} &&+ 10x^{43} &&+ 6x^{42}  &&+ 8x^{41}  &&- 5x^{40}  &&- 12x^{39} &&+ 7x^{38}  &&- 7x^{37}  &&+ 7x^{36}  \\
&+ x^{35}  &&- 3x^{34}  &&+ 10x^{33} &&+ x^{32}   &&- 6x^{31}  &&- 2x^{30}  &&- 10x^{29} &&- 3x^{28}  &&+ 2x^{27}  \\
&+ 9x^{26} &&- 3x^{25}  &&+ 14x^{24} &&- 8x^{23}  && &&- 7x^{21}  &&+ 9x^{20}  &&+ 3x^{19}  &&- 4x^{18}  \\
&- 10x^{17} &&- 7x^{16} &&+ 12x^{15} &&+ 7x^{14}  &&+ 2x^{13}  &&- 12x^{12} &&- 4x^{11}  &&- 2x^{10}  &&+ 5x^9     \\
& &&+ x^7      &&- 7x^6    &&+ 7x^5     &&- 4x^4     &&+ 12x^3    &&- 6x^2     &&+ 3x       &&- 6
\end{align}

 

Let me know if you’re enjoying these Math Bites? M x