Landau–Ramanujan constant

The Landau - Ramanujan constant is a mathematical constant, and as such belongs to number theory. Your name refers to the two famous mathematician Edmund Landau and Srinivasa Ramanujan independently chosen who demonstrated their existence. The Landau - Ramanujan constant is denoted by and has approached the Dezimalzahldarstellung K = 0.7642236535892206 ...

The investigation of the Landau - Ramanujan constant is related to the question of which natural numbers can be written as the sum of two square numbers, and the resulting problem of determining the proportion of these numbers to the natural numbers asymptotically.

Formulas

Be the number of natural numbers, which can be a positive real number represented as a sum of two square numbers. Landau and Ramanujan shown independently that asymptotically proportional to, i.e., the limit exists

Wherein represents the natural logarithm of. The limit is called the Landau - Ramanujan constant.

It still applies:

In addition, there are other formulas that the Landau - Ramanujan constant bring about in relation with the Riemann zeta function, the Dirichlet beta function, the Euler - Mascheroni constant and the lemniscate constants.

Derivation of the second equation in II

The second equation in II results from the Euler product representation of the Riemann zeta function on the half-plane. Because of you follow for using a known mathematical constant formula of Analysis:

With

And

It goes into the last equation of the above chain of equations that a prime number is either odd or equal to 2, while in the latter case either modulo 4 remainder 1 or 3.

So results

And thus

And, finally, to be displayed equation.