If you refer to the number of natural numbers that are smaller than X and can be represented as a sum of two squares, with S (x), then the following formula applies to the borderline case in which X becomes infinitely large:
K is the so-called Landau–Ramanujan constant; named after mathematicians Edmund Landau and Srinivasa Ramanujan, both of whom independently stumbled upon their existence. The numerical value of K is (approximately) 0.76422365...
Now you can ask yourself what this is supposed to do and why such a constant is good. From Landau, who demonstrated the Formula in 1908, one would probably only get a contemptuous look. He was the prime example of the type of mathematician who only accepts pure research and reject any application. Landau described practical mathematical problems as lubricating oil mathematics and its arrogance-bordering self-confidence has left traces in mathematical literature.
For Landau, only the mathematical rigor mattered: first, the required terms were defined, then a mathematical theorem was established and the proof led to it. Even today, many textbooks and lectures follow his example, consisting of endless repetitions of the structure of "definition – sentence – proof". The advantage is that you don't implicitly assume anything. But it does not necessarily make mathematics accessible. Especially the examples from applied mathematics rejected by Landau represent a didactically quite useful link between the complex abstraction of the subject and the easier-to-understand world of our everyday life.
Does Pi have to do with geometry?
Even geometry was too close to applied mathematics for Landau. He avoided using any geometric statements or examples. In one of his textbooks, for example, he defined the number π as the smallest positive zero of the sine function. And he represented the sine function in the form of an infinite series. Both are mathematically absolutely permissible and also make sense in the context of Landau's work. But it might still have been possible to mention that this number has a geometric meaning as the ratio of circumference to diameter of a circle. Or that their value is about 3.14159... Is. Or that countless people from antiquity to the present day have been busy exploring the connections between this irrational number and many natural phenomena. That would have made things clearer, but for Landau, what counted most was that the math was correct. Clarity was not a category that interested him.
Despite (or perhaps because of) this concentration on pure mathematics, many of his books, especially those dealing with prime numbers, became standard works. His name still appears today in many areas of mathematics, for example in the "Landau symbols" or the "Landau problems". The latter denote four mathematical statements that Landau enumerated at the international Congress of mathematicians in 1912. They all deal with prime numbers and have not been proven to date. One is about whether there are infinitely many primes that can be written as n2 + 1 (for integer values of n) (the so-called Landau conjecture).
Whether and when these problems are solved will be shown. The "Landau problem" of the difficult to understand mathematical texts will probably continue to exist.
