  The Galactic Center (GC), with its high density of massive stars, is a
promising target for radio transient searches. In particular, the discovery and
timing of a pulsar orbiting the central supermassive black hole (SMBH) of our
Galaxy will enable stringent strong-field tests of gravity and accurate
measurements of SMBH properties. We performed multi-epoch 4-8 GHz observations
of the inner $\approx$ 15 pc of our Galaxy using the Robert C. Byrd Green Bank
Telescope in 2019 August-September. Our investigations constitute the most
sensitive 4-8 GHz GC pulsar survey conducted to date, reaching down to a 6.1
GHz pseudo-luminosity threshold of $\approx$ 1 mJy kpc$^2$ for a pulse duty
cycle of 2.5$\%$. We searched our data in the Fourier domain for periodic
signals incorporating a constant or linearly changing line-of-sight pulsar
acceleration. We report the successful detection of the GC magnetar PSR
J1745$-$2900 in our data. Our pulsar searches yielded a non-detection of novel
periodic astrophysical emissions above a 6$\sigma$ detection threshold in
harmonic-summed power spectra. We reconcile our non-detection of GC pulsars
with inadequate sensitivity to a likely GC pulsar population dominated by
millisecond pulsars. Alternatively, close encounters with compact objects in
the dense GC environment may scatter pulsars away from the GC. The dense
central interstellar medium may also favorably produce magnetars over pulsars.
