Ramanujan summation

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent 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.

Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that:

$\begin{align} \frac{f\left( 0\right) }{2}+f\left( 1\right) +\cdots+f\left( n-1\right) + \frac{f\left( n\right) }{2} &=\frac{f\left( 0\right) +f\left( n\right) }{2}+\sum_{k=1}^{n-1}f\left(k\right)\\ &= \int_0^n f(x)\,dx + \sum_{k=1}^p\frac{B_{k+1}}{(k+1)!}\left(f^{(k)}(n)-f^{(k)}(0)\right)+R_p. \end{align}$

Ramanujan wrote it for the case p going to infinity:

$\sum_{k=1}^{x}f(k)=C+\int_0^x f(t)\,dt+\frac{1}{2}f(x)+\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(x),$

where C is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that R tends to 0 as x tends to infinity, we see that, in a general case, for functions f(x) with no divergence at x = 0:

$C(a)=\int_0^a f(t)\,dt-\frac{1}{2}f(0)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0),$

where Ramanujan assumed $a=0$. By taking $a=\infty$ we normally recover the usual summation for convergent series. For functions f(x) with no divergence at x = 1, we obtain:

$C(a)=\int_1^a f(t)\,dt+\frac{1}{2}f(1)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(1).$

C(0) was then proposed to use as the sum of the divergent sequence. It is like a bridge between summation and integration. Using standard extensions for known divergent series, he calculated "Ramanujan summation" of those. In particular, the sum of 1 + 2 + 3 + 4 + · · · is

$1+2+3+\cdots = -\frac{1}{12}\ (\Re)$

where the notation $(\Re)$ indicates Ramanujan summation. This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it was a Ramanujan summation.

For even powers we have:

$1+2^{2k}+3^{2k}+\cdots = 0\ (\Re)$

and for odd powers we have a relation with the Bernoulli numbers:

$1+2^{2k-1}+3^{2k-1}+\cdots = -\frac{B_{2k}}{2k}\ (\Re).$

Those values are consistent with the Riemann zeta function.

More recently, the use of C(1) has been proposed as Ramanujan's summation, since then it can be assured that one series $\sum_{k=1}^{\infty}f(n)$ admits one and only one Ramanujan's summation, defined as the value in 1 of the only solution of the difference equation $R(x)-R(x+1)=f(x)$ that verifies the condition $\int_1^2 R(t)\,dt=0$.

This new definition of Ramanujan's summation (denoted as $\sum_{n \ge 1}^{\Re} f(n)$) does not coincide with the earlier defined Ramanujan's summation (C(0)) nor with the summation of convergent series, but it has interesting properties, such as: If R(x) tends to a finite limit when x → +1, then the series $\sum_{n \ge 1}^{\Re} f(n)$ is convergent, and we have

$\sum_{n \ge 1}^{\Re} f(n)=\sum_{n \ge 1}^{\infty}f(n)-\lim_{n \to \infty}\int_1^n f(t)\,dt.$

Another interesting result is this:

$\sum_{n \ge 1}^{\Re} \frac{1}{n}=\gamma,$

where $\gamma$ is the Euler–Mascheroni constant.

Ramanujan resummation can be extended to integrals for example using the Euler-Maclaurin summation formula one can write

$I(n, \Lambda) = \frac{n}{2}I(n-1, \Lambda) + \zeta(-n) - \sum_{r=1}^{\infty}\frac{B_{2r}}{(2r)!} a_{n,r}(n-2r+1) I(n-2r, \Lambda)$,

$a_{n,r}= \frac{\Gamma(n+1)}{\Gamma (n-2r+2)}$

here $I(n, \Lambda ) = \int_{0}^{\Lambda}dxx^{n}$ with $\Lambda \rightarrow \infty$ the application of this Ramanujan resummation lends to finite results in the renormalization of Quantum Field theories

See also

• Borel summation
• Cesàro summation
• Ramanujan's sum
• Divergent series