Numerators of logarithms are negated sums of powers roots of unity.

(*start*)
Clear[x, roots, n, k];
TableForm[Table[polynomial = Sum[x^(k), {k, 1, n}];
  polynomial, {n, 1, 12}]]

TableForm[Table[polynomial = Sum[x^(k), {k, 1, n}];
  roots = x /. NSolve[polynomial == 0, x];
  Re[Table[-Total[roots^h], {h, 1, 12}]], {n, 1, 12}]]
(*end*)

"Numerators of Dirichlet series for logarithms are negated sums of powers roots of unity."