A solution to an open problem on the Euler–Mascheroni constant

The classical Le Verrier algorithm for the calculation of the characteristic polynomial of a matrix has led us to a solution of an open problem concerning the Euler–Mascheroni constant γ  . More exactly, we prove that there exists uniquely a sequence of rational numbers (ℓn)n⩾1(ℓn)n⩾1 such that, for any integer q⩾1q⩾1, the sequence∑k=1n1k-logn-log1+ℓ1n+⋯+ℓqnq,is an n-q-1n-q-1 order approximation of γ.