These relationships can be expressed using algebra. Then multiply this equation by 4 and subtract the second equation from the first:.
Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods.
For an extreme example, appending a single zero to the front of the series can lead to inconsistent results.
One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.
The implementation of this strategy is called zeta function regularization. The latter series is an example of a Dirichlet series. The benefit of introducing the Riemann zeta function is that it can be defined for other values of s by analytic continuation.
The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics.
Where both Dirichlet series converge, one has the identities:. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis , and Ramanujan summation, with its shortcut to the Euler—Maclaurin formula.
Instead, the method operates directly on conservative transformations of the series, using methods from real analysis.
The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows.
For convenience, one may require that f is smooth , bounded , and compactly supported. The constant term of the asymptotic expansion does not depend on f: Ramanujan wrote in his second letter to G.
Hardy , dated 27 February Ramanujan summation is a method to isolate the constant term in the Euler—Maclaurin formula for the partial sums of a series.
To avoid inconsistencies, the modern theory of Ramanujan summation requires that f is "regular" in the sense that the higher-order derivatives of f decay quickly enough for the remainder terms in the Euler—Maclaurin formula to tend to 0.
Ramanujan tacitly assumed this property. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series.
Stable means that adding a term to the beginning of the series increases the sum by the same amount. This can be seen as follows. By linearity, one may subtract the second equation from the first subtracting each component of the second line from the first line in columns to give.
In bosonic string theory , the attempt is to compute the possible energy levels of a string, in particular the lowest energy level.
Ultimately it is this fact, combined with the Goddard—Thorn theorem , which leads to bosonic string theory failing to be consistent in dimensions other than The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion.
A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.
As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: In January , Numberphile produced a YouTube video on the series, which gathered over 1.
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