- The Riemann zeta function is defined, for as

.

Moreover, it has an integral representation in terms of Euler’s gamma function, . It can actually be extended to a meromorphic function on the whole complex plane with a simple pole at .

Although, it is undoubtebly the most important function in mathematics, the Riemann zeta function still keeps many misteries. The most important and impenetrable of them is the Riemann hypothesis (*all non-trivial zeroes of the lie on the line *) with tremendous consequences in number theory and beyond!

Let us recall what happens when we evaluate at integers. First, let us start with Euler’s result from 1734 which asserts that

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Later, in 1740, the same Euler proved the following generalization:

where are the Bernoulli numbers and they are given by the Taylor series expansion . More about Euler and the zeta values can be found in this paper.

Let us remark that the key ingredient in the classical proof of Euler’s formula is the following cotangent identity which is also due to Euler:

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Expanding the quotient inside the sum sign as a geometric series and interchanging the order of summation, we obtain the following identity:

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**Remarks.**

- Euler’s formula implies the following equality of subrings of :

.

- Thanks to the functional equation

,

one can deduce the values of at negative integers: for all . In particular, for all ; we call these values the trivial zeros of the function . Also, we have and .

**Question: ***What can we say about when is odd?*

Unfortunately, not too much is known, We cannot even find a closed formula for in terms of . This led to the following:

**CONJECTURE. (Transcedence conjecture)**. *The numbers are algebraically independent, that is, for each and each nonzero polynomial , we have*

*. *

At this point, this conjecture seems completely out of reach! In 1882, Lindemann proved that is transcendental. If we combine this with Euler’s formula, it follows that are also transcendental. At this point, we do not know whether is transcendental or not. We only know that is irattional. This result was proved by Roger Apery back in 1979.

In 2001, Ball and Rivoal proved that there exist infinitely many irrational numbers among . In fact, they proved even more:

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In 2002, Wadim Zudilin proved that *at least one of the four numbers * and is irrational. Recently, Francis Brown suggested a common geometric framework for these irrationality proofs. His approach is not elementary and it is based on the study of periods of the moduli of the curves of genus zero with marked points!

2. Double zeta values. Back in the days, Euler investigated the algebraic structure of these numbers in order to find possible relations among zeta values.

We start by multiplying two Riemann zeta values:

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We call the first two terms in the last line the *double zeta values. * They admit various representations and we have

Therefore, we obtained the following identity which is also called Euler’s *reflection formula*:

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In the particular case we have or which gives us . Also, is a rational multiple of . This is nothing else than just a baby example of what we shall call *stuffle product*. Moreover, in 1775, in his attempt in finding a formula for what we call nowadays Apery’s constant, , the same Euler produced the following remarkable identity:

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Another important identity for the double zeta values is the one proved by Hoffman back in 1992,

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In 2011, Pierre Cartier and Don Zagier proved a sort of a converse of the stuffle product identity for double zeta values,

where and is odd.

3. Multiple zeta values came out as revitalization of the zeta and double zeta values 200 years later. In 1992, Michael Hoffman and Don Zagier generalized the concept of Riemann zeta function & values of the zeta function to the multiple zeta function and multiple zeta values. These objects which are also called Euler-Zagier sums are defined as

where are positive integers and . We call the above number a *multiple zeta value *of *depth* and *weight* . The MZVs can be also rewritten s the nested sums

.

Clearly, and there are MZVs of weight and depth and altogether of given weight $latex k_{1}+k_{2}+\ldots k_{r}$. The strange thing is that the vector space of the MZVs spanned over turns out to have smaller dimension!

The main goal of this theory is to understand all linear relations over among the MZVs of a given weight. Despite the fact that they look very simple, this theory of the MZVs has deep connection with many different topics such as knot invariants, Galois representations, periods of mixed Tate motives, Feyman diagrams in quantum field theory. For convenience, we have and .