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where the a k ∈ C, is called a Ramanujan expansion [12] of f (n). Ramanujan found expansions of some of the well-known functions of number theory. All of these results are proved in an "elementary" manner (i.e. only using formal manipulations of series and the simplest results about convergence).
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences.
Srinivasa Ramanujan Aiyangar [a] (22 December 1887 – 26 April 1920) was an Indian mathematician.Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then ...
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1729 can be expressed as a sum of two positive cubes in two ways, illustrated geometrically. 1729 is also known as Ramanujan number or Hardy–Ramanujan number , named after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan who was ill in a hospital.
The Ramanujan Journal; Ramanujan Math Park; Ramanujan Mathematical Society; Ramanujan prime; Ramanujan summation; Ramanujan tau function; Ramanujan theta function; Ramanujan–Nagell equation; Ramanujan–Petersson conjecture; Ramanujan–Sato series; Ramanujan–Soldner constant; Ramanujan's congruences; Ramanujan's lost notebook; Ramanujan's ...
He is a coordinating editor of The Ramanujan Journal and, in 1996, received an expository Steele Prize from the American Mathematical Society for his work editing Ramanujan's Notebooks. [ 3 ] [ 4 ] A Lester R. Ford Award was given to Berndt, with Gert Almkvist, in 1989 [ 5 ] and to Berndt, with S. Bhargava, in 1994.