On the asymptotically uniform distribution of the zeros of the partial sums of the Riemann zeta function

For every integer n ≥ 2 , let S ( n ) = { z : a ( n ) ≤ Re z ≤ b ( n ) } be the critical strip where all the zeros of the n th partial sum of the Riemann zeta function, ζ n ( z ) = ∑ k = 1 n 1 k z , are located. This paper shows that there exists N such that for n > N the set { Re z : ζ n ( z ) = 0 } is dense in the interval [ a ( n ) , b ( n ) ] . That means that every ζ n ( z ) possesses zeros near every vertical line contained in S ( n ) , provided that n > N .