  We present the data analysis pipeline, commissioning observations and initial
results from the GREENBURST fast radio burst (FRB) detection system on the
Robert C. Byrd Green Bank Telescope (GBT) previously described by Surnis et al.
which uses the 21~cm receiver observing commensally with other projects. The
pipeline makes use of a state-of-the-art deep learning classifier to winnow
down the very large number of false positive single-pulse candidates that
mostly result from radio frequency interference. In our observations totalling
156.5 days so far, we have detected individual pulses from 20 known radio
pulsars which provide an excellent verification of the system performance. We
also demonstrate, through blind injection analyses, that our pipeline is
complete down to a signal-to-noise threshold of 12. Depending on the observing
mode, this translates to peak flux sensitivities in the range 0.14--0.89~Jy.
Although no FRBs have been detected to date, we have used our results to update
the analysis of Lawrence et al. to constrain the FRB all-sky rate to be
$1140^{+200}_{-180}$ per day above a peak flux density of 1~Jy. We also
constrain the source count index $\alpha=0.83\pm0.06$ which indicates that the
source count distribution is substantially flatter than expected from a
Euclidean distribution of standard candles (where $\alpha=1.5$). We discuss
this result in the context of the FRB redshift and luminosity distributions.
Finally, we make predictions for detection rates with GREENBURST, as well as
other ongoing and planned FRB experiments.
