Exercise 16, § 1

For every complex number $s$ such that $\mathscr{R}(s)>1$, put

$n\geqslant 1$ (“Riemann zeta function”).

a) The function $\zeta$ is holomorphic on the open subset of $\mathbf{C}$ formed by the complex numbers $s$ such that $\mathscr{R}(s)>1$.

b) For $y\in \mathbf{R}_+^*$, put

this series converges absolutely and uniformly on the compact subsets of $\mathbf{R}_+^*$. For every $s$ such that $\mathscr{R}(s)>1$, one has

(see FVR, VII, p. 10 for the gamma function in the complex plane).

c) For every $y\in \mathbf{R}_+$, one has $\theta (y$ ) = $y\theta (y)$. (Use the Poisson formula and exercise 1.)

d) For every $s$ such that $\mathscr{R}(s)>1$, one has

e) There exists a unique holomorphic function on $\mathbf{C}-\{1\}$ whose restriction to the open set of complex numbers with real part $>1$ coincides with $\zeta$. This function, still denoted by $\zeta$, satisfies $(s-1)\zeta (s)\rightarrow 1$ when $s\rightarrow 1$ (in other words, it has a simple pole of residue 1 at $s= 1$).

f) The function Λ defined by

is holomorphic on $\mathbf{C}-\{0,1\}$; it has simple poles of residue 1 at $s= 0$ and $s= 1$. One has $\Lambda (1-s) = \Lambda (s)$ for every $s\in \mathbf{C}-\{0,1\}$.

g) Now prove that for every $s\in \mathbf{C}$ such that $\mathscr{R}(s)>1$, one has

where the product is over the set of prime numbers and is absolutely convergent (“Euler product”).

h) When $\sigma >1$ tends to 1, one has

$\sum p^{-\sigma}=$ log$(\frac{1}{\sigma- 1})+ O(1)$,

$p$

where the sum is over the prime numbers.

i) Every $s\in \mathbf{C}-\{1\}$ such that $\zeta (s) = 0$ is either of the form $s=-2k$, where $k\geqslant 1$ is an integer, or satisfies $0\leqslant \mathscr{R}(s)\leqslant 1$.

j) One has $\zeta (s)\not= 0$ if $\mathscr{R}(s) = 1$ and $s\not= 1$. (Prove using the Euler product that

for every $\sigma >1$ and $t\in \mathbf{R}$, then consider $t$ such that $\zeta (1 +it) = 0$.)