Approximations by special values of the Riemann zeta function via cotangent integrals

 In this talk, we will prove some approximation results by using a surprising cotangent integral identity which involves the ratio $\frac{\zeta(2k+1)}{\pi^{2k+1}}$. This cotangent integral is more flexible in controlling coefficients of zeta values compared to the one developed by Alkan (Proc. Amer. Math. Soc. 143 (9) 2015, 3743–3752.). Let $A$ be a sufficiently dense subset of $\{\zeta(3),\zeta(5),\zeta(7)\dots\}$. We show that real numbers can be approximated by certain linear combinations of elements in $A$, where the coefficients are values of the derivatives of rational polynomials. This is a joint work with Dongsheng Wu.