Showing posts from November, 2016

Is the Riemann hypothesis true?

In 2001, Jeffrey Lagarias proved that the Riemann hypothesis is equivalent to the following statement (proof here):

`\sigma(n) <= \H_n + \ln(\H_n)*\exp(\H_n)`
with strict inequality for `n > 1`, where `\sigma(n)` is the sum of the positive divisors of `n`.

In 1913, Grönwall showed that the asymptotic growth rate of the sigma function can be expressed by:

`\lim_{n to \infty}\frac{\sigma(n)}{\n \ln\ln n} = \exp(\gamma)`

where lim is the limit superior.

Relying on this two theorems, we can show that:

`lim_{n to \infty}\frac{\exp(\gamma) * n \ln \ln n}{\H_n + \ln(\H_n) * \exp(\H_n)} = 1`

with strict inequality for each `1 < n < \infty` (see Wolfram|Alpha):
`\exp(\gamma) * n \ln \ln n < \H_n + \ln(\H_n) * \exp(\H_n)`

If the Riemann hypothesis is true, then for each `n ≥ 5041`:

`\sigma(n) <= \exp(\gamma) * n \ln \ln n`
By using the usual definition of the `\gamma` constant:
`\gamma = \lim_{n to \infty}(\H_n - \ln n)`

we can reformulate the result as (see Wolfram|Alpha):

Euler-Mascheroni constant

In this post we're going to take a look at a mysterious mathematical constant, called the Euler–Mascheroni constant, and its fascinating role in harmonic and prime numbers.

# Is `gamma` transcendental? This constant, although it has a fairly simple definition, it is currently not known whether it is rational or irrational, but it is widely believed by mathematicians to be transcendental, which also implies that it is irrational.
It is usually defined as: 
`\gamma = \lim_{n to \infty}(\H_n - \ln n)`
where `\H_n` is the `n`thharmonic number, which is defined as:
`\H_n = \sum_{k=1}^(n)\frac{1}{k}`
# "Proving" that `\gamma` is transcendental There exists a proof that `\gamma` is transcendental, but the proof is very subtle:

By Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number.The `n`th harmonic number is rational.
As all rational numbers are algebraic numbers, the definition of `\gamma` reduces to: