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Home » Uncategorized » An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations

An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations

  1. The Riemann zeta function is defined, for s\in\mathbb{C} as

       \displaystyle\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}, Res>1.

 Moreover, it has an integral representation in terms of Euler’s gamma function, \displaystyle\zeta(s)=\frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}}{e^x-1}dx. It can actually be extended to a meromorphic function on the whole complex plane with a simple pole at s=1.

 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 \zeta(s) lie on the line Res=\frac{1}{2}) with tremendous consequences in number theory and beyond!

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

\displaystyle \zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.

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

\displaystyle \zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}=(-1)^{k+1}\frac{2^{2k-1}B_{2k}}{(2k)!}\cdot\pi^{2k},

where B_{k} are the Bernoulli numbers and they are given by the Taylor series expansion \displaystyle\frac{z}{e^z-1}=\sum_{k=0}^{\infty}\frac{B_{k}}{k!}\cdot z^k, |z|<2\pi. 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:

\displaystyle \pi\cot(\pi x)=\frac{1}{x}+\sum_{n\geq 1}\frac{2x}{x^2-n^2}.

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

\displaystyle \pi\cot(\pi x)=\frac{1}{x}-2\sum_{k\geq 1}\zeta(2k)x^{2k-1}.


  •  Euler’s formula implies the following equality of subrings of \mathbb{R}:

            \displaystyle\mathbb{Q}[\zeta(2), \zeta(4), \ldots]=\mathbb{Q}[\pi^2].

  •   Thanks to the functional equation

            \displaystyle \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s),

one can deduce the values of \zeta(s) at negative integers: \zeta(-k)=-\frac{B_{k+1}}{k+1} for all k\geq 1. In particular, \zeta(-2k)=0 for all k\geq 1; we call these values the trivial zeros of the function \zeta. Also, we have \zeta(0)=-\frac{1}{2} and \zeta(-1)=-\frac{1}{12}.

Question: What can we say about \zeta(s) when s is odd?

Unfortunately, not too much is known, We cannot even find a closed formula for \zeta(2n+1) in terms of \pi. This led to the following:

CONJECTURE. (Transcedence conjecture). The numbers \pi, \zeta(3), \zeta(5), \ldots are algebraically independent, that is, for each k\geq 0 and each nonzero polynomial P\in\mathbb{Z}[x_{0}, x_{1}, \ldots, x_{k}], we have

\displaystyle P(\pi, \zeta(3), \zeta(5), \ldots, \zeta(2k+1))\neq 0.  

 At this point, this conjecture seems completely out of reach! In 1882, Lindemann proved that \pi is transcendental. If we combine this with Euler’s formula, it follows that \zeta(2k) are also transcendental. At this point, we do not know whether \zeta(3) is transcendental or not. We only know that \zeta(3) 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 \zeta(2k+1). In fact, they proved even more:

\displaystyle\dim_{\mathbb{Q}}<1, \zeta(3), \zeta(5), \ldots, \zeta(2n+1)>\geq\frac{1}{3}\log n.

In 2002, Wadim Zudilin proved that at least one of the four numbers \zeta(5), \zeta(7), \zeta(9) and \zeta(11) 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 M_{0, n} of the curves of genus zero with n 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:

\displaystyle\zeta(k_{1})\zeta(k_{2})=\left(\sum_{n_{1}=1}^{\infty}\frac{1}{n_{1}^{k_{1}}}\right)\left(\sum_{n_{2}=1}^{\infty}\frac{1}{n_{2}^{k_{2}}}\right)=\sum_{ 1\leq n_{1}<n_{2}}\frac{1}{n_{1}^{k_{1}}n_{2}^{k_{2}}}+\sum_{  n_{1}>n_{2}\geq 1}\frac{1}{n_{1}^{k_{1}}n_{2}^{k_{2}}}+\sum_{ n=n_{1}=n_{2}\geq 1}\frac{1}{n^{k_{1}+k_{2}}}.

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

\displaystyle\zeta(k_{1}, k_{2})=\sum_{1\leq n_{1}<n_{2}}\frac{1}{n_{1}^{k_{1}}n_{2}^{k_{2}}}=\sum_{n\geq 2}\frac{1}{n^{k_{2}}}\left(\frac{1}{1^{k_{1}}}+\frac{1}{2^{k_{1}}}+\ldots+\frac{1}{(n-1)^{k_{1}}}\right)=\sum_{m, n\geq 1}\frac{1}{(m+n)^{k_{2}}n^{k_{1}}}.

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

\displaystyle \zeta(k_{1})\zeta(k_{2})=\zeta(k_{1}, k_{2})+\zeta(k_{2}, k_{1})+\zeta(k_{1}+k_{2}).

In the particular case k_{1}=k_{2}=2 we have \displaystyle \zeta(2)\cdot\zeta(2)=\zeta(2, 2)+\zeta(2, 2)+\zeta(4) or \zeta^2(2)=2\zeta(2, 2)+\zeta(4) which gives us \displaystyle \zeta(2, 2)=\frac{\pi^4}{120}. Also, \displaystyle \zeta(2k, 2k) is a rational multiple of \pi^{4k}. 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, \zeta(3), the same Euler produced the following remarkable identity:

\displaystyle \zeta(1, 2)=\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n-1}\frac{1}{(m-1)}\right)\frac{1}{n^2}=\sum_{n=1}^{\infty}\frac{1}{n^3}=\zeta(3).

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

\displaystyle\zeta(n+1)=\sum_{i=1}^{n-1}\zeta(i, n+1-i), n\geq 2.

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

\displaystyle \zeta(k_{1}, k_{2})=\frac{1}{2}(1+(-1)^{k_{1}})\zeta(k_{1})\zeta(k_{2})+\frac{1}{2}\left[(-1)^{k_{1}}\binom{k_{1}+k_{2}}{k_{2}}-1\right]\zeta(k_{1}+k_{2})-(-1)^{k_{1}}\sum_{i=1}^{(k_{1}+k_{2}-3)/2}\left[\binom{k_{1}+k_{2}-2i}{k_{2}-1}+\binom{k_{1}+k_{2}-2i}{k_{1}-1}\right]\cdot\zeta(2i)\cdot\zeta(k_{1}+k_{2}-2i),

where k_{1}, k_{2}\geq 1 and k_{1}+k_{2} 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

\displaystyle\zeta(k_{1}, k_{2}, \ldots, k_{r})=\sum_{1\leq n_{1}<n_{2}<\ldots<n_{r}}\frac{1}{n_{1}^{k_{1}}n_{2}^{k_{2}}\ldots n_{r}^{k_{r}}}\in\mathbb{R},

where k_{1}, k_{2}, \ldots, k_{r} are positive integers and k_{r}\geq 2. We call the above number a multiple zeta value of depth r and weight k=k_{1}+k_{2}+\ldots +k_{r}. The MZVs can be also rewritten s the nested sums

\displaystyle\zeta(k_{1}, k_{2}, \ldots, k_{r})=\sum_{n_{r}=1}^{\infty}\frac{1}{n_{r}^{k_{r}}}\sum_{n_{r-1}=1}^{n_{r}-1}\frac{1}{n_{r-1}^{k_{r-1}}}\ldots\sum_{n_{2}=1}^{n_{3}-1}\frac{1}{n_{2}^{k_{2}}}\sum_{n_{1}=1}^{n_{2}-1}\frac{1}{n_{1}^{k_{1}}}.

Clearly, 0<r<k_{1}+k_{2}+\ldots k_{r} and there are \binom{k_{1}+k_{2}+\ldots +k_{r}-2}{r-1} MZVs of weight \displaystyle k_{1}+k_{2}+\ldots k_{r} and depth r and 2^{k-2} 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 \mathbb{Q} turns out to have smaller dimension!

 The main goal of this theory is to understand all linear relations over \mathbb{Q} 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 \zeta(\{1\}^{3}, 5)=\zeta(1, 1, 1, 5) and \zeta(\{2\}^4, 6)=\zeta(2, 2, 2, 2, 6).


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