  The high stellar density in the central parsecs around the Galactic Centre
makes it a seemingly favourable environment for finding relativistic binary
pulsars. These include pulsars orbiting other neutron stars, stellar-mass black
holes or the central supermassive black hole, Sagittarius A*. Here we present
multi-epoch pulsar searches of the Galactic Centre at four observing
frequencies, (4.85, 8.35, 14.6 18.95) GHz, using the Effelsberg 100-m radio
telescope. Observations were conducted one year prior to the discovery of, and
during monitoring observations of, the Galactic Centre magnetar PSR J1745-2900.
Our data analysis features acceleration searches on progressively shorter time
series to maintain sensitivity to relativistic binary pulsars. The multi-epoch
observations increase the likelihood of discovering transient or nulling
pulsars, or ensure orbital phases are observed at which acceleration search
methods work optimally. In ~147 h of separate observations, no previously
undiscovered pulsars have been detected. Through calibration observations, we
conclude this might be due to insufficient instantaneous sensitivity; caused by
the intense continuum emission from the Galactic Centre, its large distance
and, at higher frequencies, the aggregate effect of steep pulsar spectral
indices and atmospheric contributions to the system temperature. Additionally
we find that for millisecond pulsars in wide circular orbits ~<800 d around
Sagittarius A*, linear acceleration effects cannot be corrected in deep
observations (9 h) with existing software tools. Pulsar searches of the
Galactic Centre with the next generation of radio telescopes - such as MeerKat,
ngVLA and SKA1-mid - will have improved chances of uncovering this elusive
population.
