two series arising from the alternating zeta function

The terms of the series defining the alternating zeta function

$$\eta (s):=\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n^s}\mathit{\hspace{1em}\hspace{1em}}(\text{Re}s>0),$$

a.k.a. the Dirichlet eta function, may be split into their real and imaginary parts:

$$\frac{1}{n^s}=\frac{e^{-ib\ln n}}{n^a}=\frac{\cos (b\ln n)}{n^a}-\frac{i\sin (b\ln n)}{n^a}$$

Here,  $s=a+ib$  with real $a$ and $b$.  It follows the equation

$\eta (s)=-{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{n^a}}\cos (b\ln n)+i{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^n}{n^a}}\sin (b\ln n)$(1)

containing two Dirichlet series.

The alternating zeta function and the Riemann zeta function are connected by the relation

$$\zeta (s)=\frac{\eta (s)}{1-2^{1-s}}$$

(see the parent entry (http://planetmath.org/AnalyticContinuationOfRiemannZetaToCriticalStrip)).  The following conjecture concerning the above real part series and imaginary part series of (1) has been proved by Sondow [1] to be equivalent with the Riemann hypothesis.

Conjecture.  If the equations

$$\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{n^a}\cos (b\ln n)=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{n^a}\sin (b\ln n)=\mathrm{\hspace{0.33em}0}$$

are true for some pair of real numbers $a$ and $b$, then

$$a=\mathrm{\hspace{0.33em}1}/2\mathit{\hspace{1em}\hspace{1em}}\text{or}\mathit{\hspace{1em}\hspace{1em}}a=\mathrm{\hspace{0.33em}1}.$$