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In mathematics and the field of
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, the Landau–Ramanujan constant is the positive real number ''b'' that occurs in a theorem proved by
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...
in 1908, stating that for large x, the number of
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s below x that are the sum of two
square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...
s behaves asymptotically as :\dfrac. This constant ''b'' was rediscovered in 1913 by
Srinivasa Ramanujan Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ...
, in the first letter he wrote to
G.H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
.S. Ramanujan, letter to
G.H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
, 16 January, 1913; see: P. Moree and J. Cazaran, ''On a claim of Ramanujan in his first letter to Hardy'', Exposition. Math. 17 (1999), no.4, 289-311.


Sums of two squares

By the sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
congruent to 3 mod 4 appears with an even exponent in their
prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...
. For instance, ''45 = 9 + 36'' is a sum of two squares; in its prime factorization, 32 × 5, the prime 3 appears with an even exponent, and the prime 5 is congruent to 1 mod 4, so its exponent can be odd. Landau's theorem states that if N(x) is the number of positive integers less than x that are the sum of two squares, then :\lim_\ \left(\dfrac\right)=b\approx 0.764223653589220662990698731250092328116790541 , where b is the Landau–Ramanujan constant.


History

This constant was stated by Landau in the limit form above; Ramanujan instead approximated N(x) as an integral, with the same constant of proportionality, and with a slowly growing error term.


References

{{DEFAULTSORT:Landau-Ramanujan constant Additive number theory Analytic number theory Mathematical constants Srinivasa Ramanujan