Asymptotic formulas for Vasyunin cotangent sums

•We provide an asymptotic formula for Vasyunin cotangent sums.•Vasyunin cotangent sums have applications in Riemann Hypothesis.•Our approach relies heavily on the Euler-Maclaurin summation formula. We study the Vasyunin-type cotangent sum c0(h/k)=−∑m=1k−1(m/k)cot(πhm/k), where h and k are positive coprime integers. This sum is related to Estermann zeta function. By applying the Euler-Maclaurin summation formula to a suitable function, we improve a previous large-k asymptotic approximation of c0(h/k). We also provide a procedure to compute an arbitrary number of terms of the approximation.