A note on the real projection of the zeros of partial sums of Riemann zeta function

Abstract

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function n (s) := P n k=1 1 k s , n > 2, is an accumulation point of the set fRe s : n (s) = 0g.

An estimate of the lower bound of the real parts of the zeros of the partial sums of the Riemann zeta function

“…for each integer n > 2 and a given real number x 0 . In order to emphasize the importance of the level curves, we recall that the sets R n , defined in (3.1), were characterized in [6] in terms of such curves as follows. By defining the real functions…”

“…Once the estimation of the bounds a (n) and b (n) has been settled, our interest is focused on the distribution of the real parts of the zeros of ζ n (z) in the interior of its critical interval a (n) , b (n) and, in this sense, we suggest to see [6,13,14,15,16]. A way of knowing the distribution of the zeros of ζ n (z) is by means of the study of the set of their real projections P ζn := {ℜz : ζ n (z) = 0} and its closure set, denoted by R (n) .…”

Zeroes of partial sums of the zeta-function

“…Within a single core, the intervals for θ 2 , θ 3 , θ 5 and θ 7 were initially 2 We believe that this condition indicates the presence of infinitely many zeroes. We are grateful to the referee for suggesting a means by which one might seek to establish this, based on [2], [4] and [12]. However, the weaker statement is sufficient for our purposes and we do not pursue this line of thought further.…”

“…3 Note that this covered a wider interval than was strictly necessary. 4 A single node of Phase III contains two 8-core Intel E5-2670 Sandybridge processors running at 2.6GHz.…”

On the closure of the real parts of the zeros of a class of exponential polynomials

“…In [2,Th. 2] it was given a characterization of R Gm(z) := f<z : G m (z) = 0g, m > 2, with G m (z) being the particular exponential polynomial (brie ‡y, e.p.)…”