Positive eta divisions

The Dirichlet eta function may be defined as

eta(z) = (1 - 2^(1-z)) zeta(z)

where zeta is the Riemann zeta function. If we set

et(x) = eta(1/2 + 2 pi i x/ln(2))

then it seems to be a fact, probably not noticed by anyone but I think I'll ask Danny Goldston, that Re(et(n)) (meaning the real part) for integers n is usually negative. When it's positive, it seems to say nothing especially good about the scale division. One for OEIS, I think.

Positive eta numbers: 18, 38, 39, 55, 64, 98, 121, 136, 144, 145, 158, 165, 167, 189, 190, 208, 209, 216, 225, 231, 243, 245, 251, 259, 266, 271, 279, 292, 299

Might be increasing in density to 50%, I suppose.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

Sort of dumb, I didn't check zeta, which turns out to have a very similar deal. Negative values of Re(zeta(1/2 + 2 pi i N/ln(2)) occur
at 18, 38, 39, 55, 75, 98, 121, 136, 144, 145, 158, 165, 167, 189, 190, etc.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> Sort of dumb, I didn't check zeta, which turns out to have a very similar deal. Negative values of Re(zeta(1/2 + 2 pi i N/ln(2)) occur
> at 18, 38, 39, 55, 75, 98, 121, 136, 144, 145, 158, 165, 167, 189, 190, etc.

I should add that this is related to a goofy thing called Gram's Law, so it's not completely unexplored territory.