Let function S(p,q) will be defined: Sum_{k>=r} zeta(p*n+q) where q=0..p-1, when p=1 than r=2, when p>1 and q=0 or q=1 then r=1, r=0 in rest of cases. S(1,0) = 1. S(2,0) = 3/4. S(2,1) = 1/4. All above was published in 1887 by Stieltjes p. 300. S(3,0) this sequence. S(3,1) see A339605. S(3,2) see A339606. S(4,0) see A256919. S(6,0) see A339529. S(6,4) = 1/12. Is open question if p = 6 is largest p such that S(p,q) is rational number. S(8,0) see A339530. For each pair of positive integers p > 1 and q <= p-1 occurs general formula on S(p,q) (one of three following cases): 1. case q=0 and p>=2 S(p,0)=Sum_{k>=2} 1/(k^p-1). 2. case q=1 and p>=2 S(p,1)=Sum_{k>=2} (1/(k*(k-1)) - (Sum_{n=0..p-2} k^n)/(k^p-1). 3. case q>=2 and p>=3 S(p,q)=Sum_{k>=2} (k^(p-q))/(k^p-1). Generally if p and q are both even numbers exists explicit formula by Pi, hypergeometric and trigonometric functions, because infinite series contain only even powers of Pi multipled by Bernoulli numbers. If (p or q) is odd or (p and q) are odd, formulas contains real part of digamma function where arguments are complex roots of unity. Up to now no explicite formulas for real parts digamma of complex argument.