A double inequality for a generalized-Euler-constant function

The rate of convergence of the sequence n↦γn(a):=∑k=0n−11a+k−lna+n−1a, a>0a>0, towards the generalized Eulerʼs constant γ(a):=limn→∞γn(a), where γ(1)γ(1) is the Euler–Mascheroni constant, is accurately estimated using the Euler–Maclaurin summation formula. The expressionγ(a)=Sn⁎(a,q)+Rn⁎(a,q) with parameters n,q∈Nn,q∈N, where Sn⁎(a,q) is a sum consisting of n+3q+2n+3q+2 summands and Rn⁎(a,q) is a remainder, is derived. The error term is estimated as|B2q|2(a+n+2)2q+1<(−1)q−1Rn⁎(a,q)<(1−4−q)|B2q|(a+n)2q+1<12qa+n(qeπ(a+n))2q, where BkBk is the kth Bernoulli coefficient. Two similar expressions are also established.