The topic of this article, the idea of attaching a ﬁnite value to divergent series, is no longer a purely mathematical exercise. These ﬁnite values of divergent series have found application in String theory and quantum ﬁeld theory (Casimir eﬀect).
The ﬁnite sum of real/complex numbers is always ﬁnite. The inﬁnite sum of real/complex numbers can be either ﬁnite or inﬁnite. For instance,
is ﬁnite and
If an inﬁnite sum has ﬁnite value it is said to converge, otherwise it is said to diverge. Divergence do not always mean it grows to . For instance,
are diverging, while
is converging. The convergence of a series is deﬁned by the convergence of its partial sum. For instance,
diverges because its partial sum
diverges, as . The geometric series
for all , because its partial sum
Note that the map is well-deﬁned between . In fact, is a holomorphic (complex diﬀerentiable) functions and, hence, analytic. The analytic function exists for all complex numbers except and, inside the unit disk , is same as the power series in (1). Thus, may be seen as an analytic continuation of
outside the unit disk, except at . A famous unique continuation result from complex analysis says that the analytic continuation is unique. Thus, one may consider the value of , outside the unit disk, as an ‘extension’ of the divergent series
In this sense, setting in , the divergent series may be seen as taking the ﬁnite value
Similarly, setting in , the oscillating divergent series may be seen as taking the ﬁnite value
This sum coincides with the Cesàro sum which is a special kind of convergence for series which do not diverge to . For instance,
diverges in Cesàro sum too. The Cesàro sum of a series
is deﬁned as the
Observe that the sequence
Since is not deﬁned on , we have not assigned any ﬁnite value to the divergent series
The Riemann zeta function
converges for with . Note that for , , where is the real logarithm. If then and since . Therefore, the inﬁnite series converges for all such that (i.e. ) because it is known that converges for all and diverges for all .
The Riemann zeta function is a special case of the Dirichlet series
with for all . Note that for , we get the Basel series
and called the Apéry’s constant. In fact, the following result of Euler gives the value of Riemann zeta function for all even positive integers.
If is well-deﬁned for then is also well-deﬁned for . Therefore,
Observe that is also a Dirichlet series. It can be shown that is analytic for all and . Thus, we have analytically continued for all and . There is a little work to be done on the zeroes of but is ﬁxable. In the strip , called the critical strip, the Riemann zeta function satisﬁes the relation
This relation is used to extend the Riemann zeta function to with non-positive real part, thus, extending to all complex number . Setting in the relation yields
Since is extension of the series
one may think of as the ﬁnite value corresponding to . The value of is obtained limiting process because it involves which is not deﬁned. However, there is an equivalent formulation of Riemann zeta function for all as
with a simple pole and residue at .
Recall that the analytic continuation of the geometric series is not deﬁned for and, hence, we could not assign a ﬁnite value to
But, setting above,
where is the Kronecker delta deﬁned as
The harmonic series
corresponds to which is not deﬁned. The closest one can conclude about is that
where is the Euler-Mascheroni constant which has the value .