An explicit formula for the prime counting function
AAN EXPLICIT FORMULA FOR THE PRIME COUNTING FUNCTION
KONSTANTINOS GAITANAS
Abstract.
This paper studies the behaviour of the prime counting function at some certainpoints.We show that there is an exact formula for π ( n ) which is valid for infinitely many naturalsnumbers n . Introduction
The prime counting function is at the center of mathematical research for centuries andmany asymptotic distributions of π ( n ) are well known.Many formulas have been discovered by mathematicians [1] but almost all of them are usingall the prime numbers not greater than n to calculate π ( n ). In this paper we give an exactformula which holds when a standard condition is satisfied.2. Some basic theorems
Theorem 2.1.
Let π ( n ) be the number of primes not greater than n and n ≥ .Then nπ ( n ) takes on every integer value greater than .Proof. The proof is presented in [2]. It uses only the fact that π ( N ) = o ( N ) and π ( N +1) − π ( N )is 0 or 1.We can conclude from this theorem that nπ ( n ) is an integer infinitelly often and we will use thisfact in order to prove the existence of our formula. (cid:3) Theorem 2.2. nlnn − < π ( n ) < nlnn − for every n ≥ .Proof. This is a theorem proved by J. Barkley Rosser and Lowell Schoenfeld and the proof ispresented at [3]. (cid:3) The formula for π ( n ) Theorem 3.1.
For infinitely many natural numbers n the following formula is valid: π ( n ) = n (cid:106) lnn − (cid:107) Proof.
We will make use of the above mentioned inequality in order to prove our formula.We have nlnn − < π ( n ) < nlnn − for every n ≥ n . We can see that the inequality now has the form: lnn − < nπ ( n ) < lnn − . a r X i v : . [ m a t h . N T ] N ov o nπ ( n ) lies between two real numbers a − a with a = lnn − .This means that for every n ≥
67 when nπ ( n ) is an integer we must have: nπ ( n ) = (cid:106) lnn − (cid:107) ⇔ π ( n ) = n (cid:106) lnn − (cid:107) .This completes the proof. (cid:3) We can see below at Table 1 that the formula (cid:106) nlnn − (cid:107) gives exactly the value of π ( n ) forevery natural number 67 ≤ n < nπ ( n ) being an integer[4]. n π ( n ) (cid:106) nlnn − (cid:107)
96 24 24100 25 25120 30 30330 66 66335 67 67340 68 68350 70 70355 71 71360 72 721008 168 1681080 180 1801092 182 1821116 186 1861122 187 1871128 188 1881134 189 1893059 437 4373066 438 4383073 439 4393080 440 4403087 441 4413094 442 442TABLE 1
References [1] Hardy, G. H., E. M. Wright An Introduction to the Theory of Numbers (5th Edition), Oxford, England,Clarendon Press, 1979.[2] On the Ratio of N to π ( N ) Solomon W. Golomb The American Mathematical Monthly Vol. 69, No. 1(Jan., 1962), pp. 36-37[3] J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill.Journ. Math. 6 (1962) 64-94.[4] OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences, http://oeis.org. Department of Applied Mathematical and Physical Sciences, National Technical Universityof Athens, Greece
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