Zeta function and Gamma function

1. Zeta function and Gamma function For anyp >1,the infinite series P∞

n=11/n^p is convergent byp-test. We set ζ(p) =

∞

X

n=1

1

n^p, p >1.

Then ζ : (1,∞) → R defines a function called the zeta function. Using Gamma function, we know

Z ∞

0

e^−ntt^p−1dt= Γ(p) n^p .

We can rewrite the zeta function by the following infinite series:

ζ(p) = 1 Γ(p)

∞

X

n=1

Z ∞

0

e^−ntt^p−1dt.

Assuming that we can change the order of sum and integration freely (this can be proved using real analysis), then

ζ(p) = 1 Γ(p)

Z ∞

0

t^p−1

∞

X

n=1

e^−nt

! dt.

The infinite seriesP∞

n=1e^−nt is a geometric series; hence

∞

X

n=1

e^−nt= e^−t

1−e^−t = 1 e^t−1.

Then zeta function can be expressed in terms of the following integral ζ(p) = 1

Γ(p) Z ∞

0

t^p−1 e^t−1dt.

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