## Posts

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