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.
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