Accurate double inequalities for generalized harmonic numbers

For n∈Nn∈N and p∈Rp∈R the nth harmonic number of order pH(n,p):=∑k=1n1kpis expressed in the form H(n,p)=H˜q(m,n,p)+Rq(m,n,p)where m,q∈Nm,q∈N are parameters controlling the magnitude of the error term. The function H˜q(m,n,p) consists of m+2q+1m+2q+1 simple summands and the remainder Rq(m, n, p) is estimated, for p ≥ 0, as 0≤(−1)q+1Rq(m,n,p)<1π(1−2·4−q)(p2+q−1)πm)2q−1·1mp.Similar result is obtained also for p   < 0 and for real zeta function (n=∞,n=∞,p > 1) as well.