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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.
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.
Download as PDF; Printable version; From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Ramanujan's sum; Retrieved from " ...
Later, I. M. Vinogradov extended the technique, replacing the exponential sum formulation f(z) with a finite Fourier series, so that the relevant integral I n is a Fourier coefficient. Vinogradov applied finite sums to Waring's problem in 1926, and the general trigonometric sum method became known as "the circle method of Hardy, Littlewood and ...
In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation , an equation to be solved in integers where one of the variables appears as an exponent .
Download as PDF; Printable version; ... i think Ramanujan sum is also related to circle method to evaluate (n--> oo) the integrals: ... The Titchmarsh book Theory of ...