Riemann zeta function

The **Riemann zeta function** or **Euler–Riemann zeta function**, *ζ*(*s*), is a function of a complex variable *s* that analytically continues the sum of the Dirichlet series

ζ
(
s

# n
1

∞

1

n
s

{\displaystyle \zeta(s) =\sum\_{n=1}^\infty\frac{1}{n^s}}

which only works when the real part of s is greater than 1. It diverges for everything else.

But using the techniques of analytic continuation on the equation above we can determine the values when the real part of s is less than 1. For example:

*ζ*(−1) = − 112.
But pluging a value of -1 into **Eq. 1** gives

ζ
(
−
1

# n
1

∞

n

=

{\displaystyle \zeta(-1) =\sum\_{n=1}^\infty {n} = }

 1 + 2 + 3 + 4 + ⋯
which obviously diverges. So what gives?

In reality the true equation for the Riemann zeta function which works for all values of s is given by

ζ
(
s
# )

1

Γ
(
s
)

∫
0
∞

x

s
−
1

e
x

−
1

d

x

{\displaystyle \zeta(s) = \frac{1}{\Gamma(s)} \int\_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x }

where

Γ
(
s
# )

∫
0
∞

x

s
−
1

e

−
x

d

x

{\displaystyle \Gamma(s) = \int\_0^\infty x^{s-1}\,e^{-x} \, \mathrm{d}x }

is the gamma function.

For the special case where the real part of s is greater than 1, *ζ*(*s*) always converges, and can be simplified to the following infinite series:

ζ
(
s

# n
1

∞

1

n

s

=

1

1
s

1

2
s

1

3
s

⋯

# σ
Re
⁡
(
s
)
>

{\displaystyle \zeta(s) = \sum\_{n=1}^\infty \frac{1}{n^{s} }= \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \quad \quad \sigma = \operatorname{Re}(s) > 1.}

That the analytic continuation of a simple formula (**Eq. 1**) could result in a more complex formula (**Eq. 2**) that just happens to reduce to the simple formula over the limited domain of the simple formula is far from obvious so I mention it here.

## Euler product formula

The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

∑

# n
1

∞

1

n
s

=

∏

p
 prime

1

1
−

p

−
s

,

{\displaystyle \sum\_{n=1}^\infty\frac{1}{n^s} = \prod\_{p \text{ prime}} \frac{1}{1-p^{-s}},}

where, by definition, the left hand side is *ζ*(*s*) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):

∏

p
 prime

1

1
−

p

−
s

=

1

1
−

2

−
s

⋅

1

1
−

3

−
s

⋅

1

1
−

5

−
s

⋅

1

1
−

7

−
s

⋅

1

1
−

11

−
s

⋯

1

1
−

p

−
s

⋯

{\displaystyle \prod\_{p \text{ prime}} \frac{1}{1-p^{-s}} = \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}} \cdots \frac{1}{1-p^{-s}} \cdots}

## Mellin transform

The Mellin transform of a function *f* is

{

M

f

}

(
s
# )
φ
(
s
# )

∫
0
∞

x

s

⋅

1
x

⋅
f
(
x
)

d
x
.

{\displaystyle \left\{\mathcal{M}f\right\}(s) = \varphi(s)=\int\_0^\infty x^{s} \cdot \frac{1}{x} \cdot f(x) \, dx.}

It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function,

ζ
(
s
)

{\displaystyle \zeta(s)}

.

Let

f
(
x
# )

1

e
x

−
1

{\displaystyle f(x)=\frac{1}{e^x-1}}

.
Then

M

f
(
s
)

=

∫
0
∞

x

s
−
1

1

e
x

−
1

d
x

=

∫
0
∞

x

s
−
1

e

−
x

1
−

e

−
x

d
x

=

∫
0
∞

x

s
−
1

∑

# n
1

∞

e

−
n
x

d
x

=

∑

# n
1

∞

∫
0
∞

x

s

e

−
n
x

d
x

x

=

∑

# n
1

∞

1

n
s

Γ
(
s
)

=
Γ
(
s
)
ζ
(
s
)

{\displaystyle
\begin{align}
\mathcal M f (s) &= \int\_0^\infty x^{s-1}\frac{1}{e^x-1}dx \\
&= \int\_0^\infty x^{s-1}\frac{e^{-x}}{1-e^{-x}}dx \\
&= \int\_0^\infty x^{s-1}\sum\_{n=1}^\infty e^{-nx}dx \\
&= \sum\_{n=1}^\infty \int\_0^\infty x^{s}e^{-nx}\frac{dx}{x} \\
&= \sum\_{n=1}^\infty \frac{1}{n^s}\Gamma(s) \\
&= \Gamma(s)\zeta(s)
\end{align}
}

Thus

ζ
(
s
# )

1

Γ
(
s
)

∫
0
∞

x

s
−
1

1

e
x

−
1

d
x
.

{\displaystyle
\zeta(s)=\frac{1}{\Gamma(s)}\int\_0^\infty x^{s-1}\frac{1}{e^x-1}dx.
}

## External links

* A simple way to look at the Riemann Zeta Function