I SHOULD HAVE written below (note negative sign on exponent of
the erroneous "pi^(1-s)/2)":
pi^(-s/2) gamma(s/2) zeta(s) = pi^(-(1-s)/2) gamma((1-s)/2) zeta(1-s).
Then we get:
zeta(1-s) = pi^(1/2-s) gamma(s/2) zeta(s) / (gamma((1-s)/2)
Now plug in s = 2:
zeta(-1) = pi^(-3/2) gamma(1) zeta(2) / gamma(-1/2)*
or
= pi^(-3/2) * 1 * (pi^2)/6 / (-2 sqrt(pi))
= pi^-2 * pi^2 /(-2 * 6)
= -1/12.
Therefore
1 + 2 + 3 + ... = -1/12 Q.E.D.
—Dan
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* since gamma(s) gamma(1-s) = pi/(sin(pi*s) implies gamma(1/2) = sqrt(pi),
and hence gamma(1/2) = (-1/2) gamma(-1/2) shows gamma(-1/2) = -2 sqrt(pi).
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How zeta(s) is eventually defined on (almost) the whole complex plane
is interesting. Let s be a complex number. Then for Re(s) > 1 the
expression
(*) zeta(z) = Sum_{n=1 to oo} 1/n^s
converges.* So zeta is now defined for the half-plane Re(s) > 1.
Now multiply both sides of (*) by (1 - 2/2^s) times itself to get
(**) (1-1/2^(s-1)) zeta(s) = Sum_{n=1 to oo} (-1)^(n+1) / n^s
This equation holds for all s with Re(s) > 1, so by permanence
it must hold for any s where zeta is defined. Luckily, the RHS of
(**) converges for a larger half-plane, namely where Re(s) > 0.
So we can solve for zeta as long as we can divide both sides by
(1 - 1/2^(s-1)), namely as long as s is unequal to 1. (Which is
just as well, since zeta has a pole as s = 1.)
zeta(s) = 1/(1-1/2^(s-1)) * Sum_{n=1 to oo} (-1)^(n+1) / n^s
for all s with Re(s) > 0 and s unequal to 1.
But there is one more trick enabling zeta to be extended to include
the left half-plane so zeta(s) is defined for all s unequal to 1. That
is the functional equation of zeta in symmetric form, namely, the
function xi given by
xi(s) = pi^(-s/2) gamma(s/2) zeta(s)
satisfies
(***) xi(s) = xi(1-s)
or written out,
pi^(-s/2) gamma(s/2) zeta(s) = pi^((1-s)/2) gamma((1-s)/2) zeta(1-s).
OOPS - sign is wrong in pi^(1-s)/2 term, which should read pi^-(1-s)/2.
A priori this can make sense only on an open set that is invariant
under the mapping s —> 1-s, but we have such an open set: the strip
{s in C | 0 < Re(s) < 1}.
But permanence implies that any functional equation for an analytic
function that holds on an open set holds everywhere. So (***) can be
used to newly define zeta anywhere that
zeta(1-s) = pi^(-s/2) gamma(s/2) zeta(s) / (pi^((1-s)/2) gamma((1-s)/2))
defines a value we hadn't defined before. Since we already know zeta for
all s unequal to 1 with Re(s) > 0, this takes care of the remaining s (Re(s) <= 0).
—Dan
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* with n^s defined as exp(s * ln(n)
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