The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where 
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch zeta-function; see e.g. [Wikipedia][1]. Below are pictures of the sets 
$\{(b,p,\Re L(b/2,1/2,p+1)/2^p)\colon\,0.03<b<5,-0.9<p<15\}$ and 
$\{(b,p,\mathrm{sgn}\,\Re L(b/2,1/2,p+1)\colon\,0.03<b<10,-0.9<p<15\}$,  
which suggest that the statement is true. This problem arises as part of the problem stated at [Is the integral always nonzero?][2]

![pics][3]


  [1]: http://en.wikipedia.org/wiki/Lerch_zeta_function
  [2]: http://mathoverflow.net/questions/204459/is-the-integral-always-nonzero/204657#204657
  [3]: https://i.sstatic.net/dgRkW.png