22 Dec On Ramanujan’s 133rd Birth Anniversary: Famous constants related to Srinivasa Ramanujan
1. Almost Integer (Ramanujan’s Constant)
Ramanujan’s constant is the transcendental number , which is an almost integer, in that it is very close to an integer:
This number was discovered in 1859 by the mathematician Charles Hermite. In a 1975 April Fools Day article in the Scientific American magazine, “Mathematical Games” columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted – hence its name. This coincidence is explained by complex multiplication and the -expansion of the -invariant.
2. Landau-Ramanujan constant
In mathematics and the field of number theory, the Landau-Ramanujan constant is the positive real number that occurs in a theorem proved by Edmund Landau in 1908, stating that for large , the number of positive integers below that are the sum of two square numbers behaves asymptotically as
This constant was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy.
Landau’s theorem states that if is the number of positive integers less than that are the sum of two squares, then
where is the Landau-Ramanujan constant.
This constant was stated by Landau in the limit form above; Ramanujan instead approximated as an integral, with the same constant of proportionality, and with a slowly growing error term.
3. Ramanujan-Soldner constant
In mathematics, the Ramanujan-Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner.
Its value is approximately .
Since the logarithmic integral is defined by
we have thus where .
4. Reciprocal of
Ramanujan is famous for his 17 series on such as:
To see more about his work on and his paper “Modular equations and approximations to ”, click on this link.
5. Sum of all natural numbers
Srinivasa Ramanujan presented two derivations of in chapter 8 of his first notebook.
The second key insight is that the alternating series is the formal power series expansion of the function but with defined as 1. Accordingly, Ramanujan writes:
. Dividing both sides by -3, one gets
Second derivation (using Ramanujan Summation):
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
For a function , the classical Ramanujan sum of the series is defined as:
where is the -th derivative of and is the -th Bernoulli number: , and so on. Setting , the first derivative of is 1, and every other term vanishes, so: .
6. Taxicab Number
The -th taxicab number, typically denoted Ta(n), also called the -th Hardy-Ramanujan number, is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in distinct ways. The most famous taxicab number is .
The name is derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan. As told by Hardy:
I remember once going to see him [Ramanujan] when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.
So far, six taxicab numbers are known.
Angad is a final year student (in 2020) of B.Tech. in Pune Institute of Computer Technology. He has found various results in the fields of pure mathematics related to Ramanujan such as real and complex analysis, analytic number theory, infinite series, approximations and other miscellaneous topics. He has published some of his results in the Romanian Mathematical Magazine and a note in The Mathematical Gazette (Nov 2020, Volume 104, Pg. 520-522).