  We report herschel observations of 100 very luminous, optically selected AGNs
at z=2-3.5 with log(LUV)(erg/sec)> 46.5, where LUV=L1350A. The distribution in
LUV is similar to the general distribution of SDSS AGNs in this redshift and
luminosity interval. We measured SF luminosity, LSF, and SFR in 34 detected
sources by fitting combined SF and WISE-based torus templates. We also obtained
statistically significant stacks for the undetected sources in two luminosity
groups. The sample properties are compared with those of very luminous AGNs at
z>4.5. The main findings are: 1) The mean and the median SFRs of the detected
sources are 1176 and 1010 Msun/yr, respectively. The mean SFR of the undetected
sources is 148 Msun/yr. The ratio of SFR to BH accretion rate is approximately
80 for the detected sources and less than 10 for the undetected sources. There
is no difference in LAGN and only a very small difference in L(torus) between
detected and undetected sources. 2) The redshift distribution of LSF and LAGN
for the most luminous, redshift 2-7 AGNs are different. The highest LAGN are
found at z=~3. However, LSF of such sources peaks at z=~5. Assuming the objects
in our sample are hosted by the most massive galaxies at those redshifts, we
find many of them are below the main-sequence of SF galaxies at z=2-3.5. 3) The
SEDs of dusty tori at high redshift are similar to those found in low redshift,
low luminosity AGNs. Herschel upper limits put strong constraints on the long
wavelength SED ruling out several earlier suggested torus templates. 4) We find
no evidence for a luminosity dependence of the torus covering factor in sources
with log(LAGN)=44-47.5. This conclusion is based on the highly uncertain and
non-uniformally treated LAGN in many earlier studies. The median covering
factors over this range are 0.68 for isotropic dust emission and 0.4 for
anisotropic emission.
