Fast Rise of "Neptune-Size" Planets ($4-8 R_{\rm Earth}$) from $P\sim10$ to $\sim250$ days -- Statistics of Kepler Planet Candidates Up to $\sim 0.75 {\rm AU}$

We infer the period ($P$) and size ($R_p$) distribution of Kepler transiting planet candidates with $R_p\ge 1 R_{\rm Earth}$ and $P < 250$ days hosted by solar-type stars. The planet detection efficiency is computed by using measured noise and the observed timespans of the light curves for $\sim 120,000$ Kepler target stars. We focus on deriving the shape of planet period and radius distribution functions. We find that for orbital period $P>10$ days, the planet frequency d$N_p$/d$\log$P for "Neptune-size" planets ($R_p = 4-8 R_{\rm Earth}$) increases with period as $\propto P^{0.7\pm0.1}$. In contrast, d$N_p$/d$\log$P for "super-Earth-size" ($2-4 R_{\rm Earth}$) as well as "Earth-size" ($1-2 R_{\rm Earth}$) planets are consistent with a nearly flat distribution as a function of period ($\propto P^{0.11\pm0.05}$ and $\propto P^{-0.10\pm0.12}$, respectively), and the normalizations are remarkably similar (within a factor of $\sim 1.5$ at $50 $ days). Planet size distribution evolves with period, and generally the relative fractions for big planets ($\sim 3-10 R_{\rm Earth}$) increase with period. The shape of the distribution function is not sensitive to changes in selection criteria of the sample. The implied nearly flat or rising planet frequency at long period appears to be in tension with the sharp decline at $\sim 100$ days in planet frequency for low mass planets (planet mass $m_p < 30 M_{\rm Earth}$) recently suggested by HARPS survey. Within $250$ days, the cumulative frequencies for Earth-size and super-Earth-size planets are remarkably similar ($\sim 28 %$ and $25%$), while Neptune-size and Jupiter-size planets are $\sim 7%$, and $\sim 3%$, respectively. A major potential uncertainty arises from the unphysical impact parameter distribution of the candidates.