# Where is the Ramanujan sum used?

## Where is the Ramanujan sum used?

Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan’s sums are used in the proof of Vinogradov’s theorem that every sufficiently-large odd number is the sum of three primes.

## What is the mistake in Ramanujan summation?

The incorrect proof The placement of that C on the end is an assumption. It assumes that this sum has a well-defined value, on which standard operations (addition, subtraction, mulitplication, division) are then defined. But this is obviously untrue.

What is used in most in Ramanujan’s theorems?

is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams). A similar result was also obtained by Glaisher.

How does Ramanujan summation work?

For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.

### Is Ramanujan summation wrong?

Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. …

### Why is Ramanujan series wrong?

Most of Ramanujan’s mistakes arise from his claims in analytic number theory, where his unrigorous methods led him astray. In particular, Ramanujan thought his approximations and asymptotic expansions were considerably more accurate than warranted.

How much is all the numbers added up to 100?

The sum of all natural numbers from 1 to 100 is 5050. The total number of natural numbers in this range is 100. So, by applying this value in the formula: S = n/2[2a + (n − 1) × d], we get S=5050.

Which is an example of a Ramanujan summation?

This formula originally appeared in one of Ramanujan’s notebooks, without any notation to indicate that it exemplified a novel method of summation. For example, the of 1 − 1 + 1 − ⋯ is: Ramanujan had calculated “sums” of known divergent series. It is important to mention that the Ramanujan sums are not the sums…

## When did Srinivasa Ramanujan write the sum theorem?

Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan’s sums are used in the proof of Vinogradov’s theorem that every sufficiently-large odd number is the sum of three primes.

## What is the Ramanujan expansion of f ( n )?

If f ( n) is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), then a convergent infinite series of the form: where the ak ∈ C, is called a Ramanujan expansion of f ( n ). Ramanujan found expansions of some of the well-known functions of number theory.

How are sums used in Vinogradov’s theorem?

Srinivasa Ramanujan mentioned the sums in a 1918 paper. In addition to the expansions discussed in this article, Ramanujan’s sums are used in the proof of Vinogradov’s theorem that every sufficiently-large odd number is the sum of three primes. is read ” a divides b ” and means that there is an integer c such that b = ac.

29/08/2019