The complex field, denoted $C$, is the set of all the complex numbers of the form $σ+it$, where $σ$ and $t$ are real numbers. Addition and multiplication over the field are defined as follows:

$(a+ib)+(c+id)(a+ib)×(c+id) =(a+c)+i(b+d)=(ac−bd)+i(ad+bc) $Over $C$, we define the real function $Re(s)$ as follows:

$Re:Ca+ib →R↦a. $For $Re(s)>1$, $ζ$ is defined as an absolutely convergent
infinite series that maps values from $C$ unto itself,
with the exception of $s=1$ where the sum diverges
and $ζ$ has a pole^{1}.

where $Γ$ is the Gamma function, a particularly useful analytic continuation of factorials to non-integral points.

$Γ(s) =0∫∞ t_{s−1}e_{−t}dt $Show that $n=1∑∞ n_{s}1 $ and $Γ(s)1 0∫∞ e_{x}−1x_{s−1} dx$ are indeed equivalent.

Here's some further context about gamma extensions that might be useful.

The zeta zeroes are what make the zeta function particularly interesting.
$ζ$ has trivial zeros^{2} at $s=−2,−4,−6,…$.
Besides these, all the other zeroes, so far, have been found to have real part $21 $.

The Riemann hypothesis, which remains unproven,
asserts that indeed **ALL** the non-trivial zeroes have $Re(s)=21 $.

Countless theories in fields as far apart as number theory, quantum mechanics, and cryptography assume that the Riemann hypothesis is true.

### Globality of Primes

In $1737$, Leonhard Euler proved that $n=1∑∞ n_{s}1 $ is equivalent to $pprime∏ 1−p_{s}1 1 $, which directly relates the zeta function to the prime numbers. For example, this result can be used to construct a direct proof of Euclid's theorem, which posits that there are infinitely many primes, by equating the two equations at $s=1$;

$ζ(1)=n=1∑∞ n1 =pprime∏ 1−p1 1 , $where $n=1∑∞ n1 $ is the harmonic series that diverges to infinity. Thus, the product must also diverge to infinity, which implies an infinite number of terms terms in the product that are greater than $1$.

Can you think of an alternative proof of Euclid's theorem?

### Euclid's Alternate Proof

Suppose there are only finitely many primes.
Let $p$ be the largest prime.
Take $N=(2×3×5×⋯×p)+1$.
Then $N−1$ is divisible by all primes less than $p$, so $N$ *must not* be divisible
by any of those primes (since it would leave a remainder of $1$).

Therefore, $N$ is either **prime** or **divisible by a prime greater than $p$**,
which contradicts the fact that we picked $p$ to be the largest possible such prime.

### Locality of Primes

The above equation gives an estimate of the global distribution of primes and them being unbounded. However, a much stronger result which directly relates to the Reimann hypothesis itself estimates the locality of primes.

Gauss^{3} posited the prime-counting function $π(x)$ as follows:

where $P_{≤x}$ is the set of all prime numbers less than $x$.
This $π$ is more commonly referred to as the *prime-counting* function.
That is, it is a step function over $R_{≥0}$ that starts at $0$
and increases by $1$ at each prime number.

Riemann developed a *prime-power counting* function
$Π_{0}(x)=21 (p_{n}<x∑ n1 +p_{n}≤x∑ n1 )$.
Riemann then showed that the zeta zeroes can be used to very accurately
approximate the locality of primes.
Here's a nice discussion of the consequences.