THE MASTER FUNCTION AND APPLICATIONS THEOPHILUS AGAMA 1 . Abstract. In this paper we introduce a function that is neither additive nor multiplicative, and is somewhat akin to the Von Mangoldt function. As an application we show that  p≤x/2 π(p) p ≥ (1 + o(1)) log log x as x −→ ∞, and  p≤x/2 θ(x/p)  log x log p − 1  -1 ≪ x log log x where p runs over the primes. 1. Introduction In analytic number theory, It is common practice to estimate the partial sums of the form  n≤x f (n)g(n) where f (n) and g(n) are arithmetic functions, which may or may not be additive or multiplicative. There are vast array of tools in the literature designed for handling such sums, especially when none of the function is trivial. In the sense that f ≡ 1 or g ≡ 1, for in such case the sum becomes somewhat easy to estimate. These sums can be hard to estimate mostly if the sum is restricted to a certain subsequence of the integers, like the primes for instance. There are, however, exceptional cases where most of the tools do not work. To make a head-way it suffices to know estimates for functions f (n) or g(n). In this short paper we prove an asymptotic inequality for the sum given by  p≤x/2 π(p) p ≥ (1 + o(1)) log log x. Date : November 1, 2018. 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. primes. 1 . 1 0DQXVFULSW &OLFN KHUH WR GRZQORDG 0DQXVFULSW MRXUQDOWH[ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65