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)`

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` th   harmonic 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 numb