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VIII – Cauchy Theory
Relations (15) and (16) resemble those proved in Chap. VII, n
◦ 28 for the
Jacobi function
θ(z) =
exp
πin
2 z
= 1 + 2
q + q
4 + q
9 + q
16 + . . .
,
where q = exp(πiz). As an aside, note that the θ(x) series used to obtain the
functional equation of the zeta function is in fact the value of θ(z) for z = ix.
Here too, the series converges for Im(z) > 0. Thanks to Poisson’s summation
formula and to the fact that the function x → exp(−πx
2 ) is equal to its
Fourier transform, we showed that
θ(−1/z) = (z/i)
1/2 θ(z) .
This is why Riemann used it to prove the functional equation for his series
ζ(s). We also have
θ(z + 2) = θ(z) .
Therefore, the function θ(z)
2 satisfies (15) and (16). In fact,
f (z) = θ(z)
2 .
(15.17)
In Chap. XII, the proof of (17) will lead us directly to the theory of modular
functions and to the classical formula giving the number of ways an integer
can be represented as the sum of two squares.