Another look at Zagier’s formula for multiple zeta values involving Hoffman elements

Abstract

In this paper, we give an elementary account into Zagier’s formula for multiple zeta values involving Hoffman elements. Our approach allows us to obtain direct proof in a special case via rational zeta series involving the coefficient $\zeta (2n)$. This formula plays an important role in proving Hoffman’s conjecture which asserts that every multiple zeta value of weight k can be expressed as a $\mathbb {Q}$-linear combinations of multiple zeta values of the same weight involving 2’s and 3’s. Also, using a similar hypergeometric argument via rational zeta series, we produce a new Zagier-type formula for the multiple special Hurwitz zeta values.

Appendix

The Clausen function (integral) is defined by

$$\begin{aligned} \displaystyle {\text {Cl}}_{2}(\theta )=-\int _0^{\theta }\log \left( 2\sin \frac{t}{2}\right) dt=\sum _{k=1}^{\infty }\frac{\sin (k\theta )}{k^2}, \end{aligned}$$

and its Taylor series expansion is given by

$$\begin{aligned} \displaystyle \frac{{\text {Cl}}_{2}(\theta )}{\theta }=1-\log |\theta |+\sum _{n=1}^{\infty }\frac{\zeta (2n)}{n(2n+1)}\left( \frac{\theta }{2\pi }\right) ^{2n}, |\theta |<2\pi . \end{aligned}$$

The higher order Clausen functions are

$$\begin{aligned} \displaystyle {\text{ Cl }}_{2m}(\theta )=\sum _{k=1}^{\infty }\frac{\sin (k\theta )}{k^{2m+1}}, {\text{ Cl }}_{2m+1}(\theta )=\sum _{k=1}^{\infty }\frac{\cos (k\theta )}{k^{2m+1}}. \end{aligned}$$

Using the properties of the Riemann zeta function, we have the following particular values:

$$\begin{aligned} \displaystyle {\text {Cl}}_{2m}(\pi )=0, {\text {Cl}}_{2m+1}(\pi )=-\frac{(4^m-1)\zeta (2m+1))}{4^m} \end{aligned}$$

and

$$\begin{aligned} \displaystyle {\text {Cl}}_{2m}\left( \frac{\pi }{2}\right) =\beta (2m), {\text {Cl}}_{2m+1}\left( \frac{\pi }{2}\right) =-\frac{(4^m-1)\zeta (2m+1)}{2^{4m+1}}, \end{aligned}$$

where $\beta (s)=\sum _{n=0}^{\infty }\frac{(-1)^n}{(2n+1)^s}, {\text {Res}}>0$ is the Dirichlet beta function.

Moreover,

$$\begin{aligned} \displaystyle \frac{d}{d\theta }{\text {Cl}}_{2m}(\theta )={\text {Cl}}_{2m-1}(\theta ), \frac{d}{d\theta }{\text {Cl}}_{2m+1}(\theta )=-{\text {Cl}}_{2m}(\theta ), \end{aligned}$$

and

$$\begin{aligned} \displaystyle \int _0^{\theta }{\text {Cl}}_{2m}(x)dx=\zeta (2m+1)-{\text {Cl}}_{2m+1}(\theta ), \int _0^{\theta }{\text {Cl}}_{2m-1}(x)dx={\text {Cl}}_{2m}(\theta ). \end{aligned}$$

Proof of the Lemma 2.6

(see also [19]). Theorem 2.4 for $z=\frac{1}{2}$ gives us

$$\begin{aligned} \displaystyle \int _0^{\frac{\pi }{2}}x^p\cot xdx= & {} \left( \frac{\pi }{2}\right) ^p\left( \log 2+\sum _{k=1}^{[p/2]}\frac{p!(-1)^k(4^k-1)}{(p-2k)!(2\pi )^{2k}}\zeta (2k+1)\right) \\&+\delta _{\left[ \frac{p}{2}\right] , \frac{p}{2}}\frac{p!(-1)^{\frac{p}{2}}\zeta (p+1)}{2^p}. \end{aligned}$$

On the other hand, since $\cot x=-2\sum \nolimits _{n=0}^{\infty }\frac{\zeta (2n)}{\pi ^{2n}}\cdot x^{2n-1}, |x|<\pi$, by integration and Fubini’s theorem, we obtain

$$\begin{aligned}&\displaystyle \int _0^{\frac{\pi }{2}}x^p\cot xdx=\int _0^{\frac{\pi }{2}}x^p\left( -2\sum _{n=0}^{\infty }\frac{\zeta (2n)}{\pi ^{2n}}\cdot x^{2n-1}\right) dx\\&\quad =-2\sum _{n=0}^{\infty }\frac{\zeta (2n)}{\pi ^{2n}}\int _0^{\frac{\pi }{2}}x^{2n+p-1}dx \end{aligned}$$

or equivalently,

$$\begin{aligned} \displaystyle \int _0^{\frac{\pi }{2}}x^p\cot xdx=-2\left( \frac{\pi }{2}\right) ^p\sum _{n=0}^{\infty }\frac{\zeta (2n)}{(2n+p)2^{2n}}, \end{aligned}$$

and the lemma follows immediately.$\square$