eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2019-01-24 7 1 1 4 10.12691/tjant-7-1-1 TJANT2019711 article Theophilus Agama theophilus@aims.edu.gh, emperordagama@yahoo.com 1 Department of Mathematics, African Institute for Mathematical Science, Ghana Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that where This is an improvement to the estimate found in the literature. http://pubs.sciepub.com/tjant/7/1/1/tjant-7-1-1.pdf prime theta