Formulae for Computing Logarithmic Integral Function ( )!

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1 Formulae for Computing Logarithmic Integral Function x 2 ln t Li(x) dt Amrik Singh Nimbran 6, Polo Road, Patna, INDIA simnimas@yahoo.co.in Abstract: The prime number theorem states that the number of primes up to a given number is approximated by the logarithmic integral function. To compute the value of this function, the author offers two formulae deduced from truncated series one convergent and the other divergent. He also gives a table of values computed by him for this function. Key words: Prime numbers, prime counting function, logarithmic integral function. AMS Classification: 11A41, 26A Introduction: Hadamard and de la Vallée Poussin independently showed in 1896 that if π(x) denotes the number of primes up to a given number x, then for some positive constant c π(x) Li(x) O(xe ). Here Li(x) is the logarithmic integral function. Some books on number theory record the values of Li(x) for certain x without giving any clue to the method by which these values have been arrived at. We find the following expansion at some places [1]: Li(x) ()! O (1) In this paper, I offer two alternative series for computing Li(x). Let us take the identity: 1!!!!! (2) Term-by-term integration of the right hand side and repeated integration by parts of the left hand side lead to: RHS ln u u (3) LHS! ()! (4) Putting u ln t, du, e t, we obtain from (2), (3) and (4): 1

2 ln ln t ln t (5)!! ()! (6) Now taking the definite integral: Li(x), we obtain two alternative expansions while (5) gives a convergent series, (6) results in a difference of two divergent series. I have found on close examination that we may usefully employ truncated versions of these expansions for computing Li(x). Let me first introduce two theorems which analyse the behaviour of the terms in (5) and (6). Theorem 1: a all x 91. attains the maximum value at n [ln x] 1 or at n [lnx] 2 for Proof: If we show that a < a and a >a at n [ln x] 1 for x 91, this will suffice. Now [ln x] (ln x) d, 0 d<1. () () {( )} {( )} and for denominator to be non-zero and positive, it is necessary that ln x > d 3, i.e., ln x 4, i.e., x 55. Now numerator denominator (d 2) (d 1) ln x. Thus () () < ln x which implies that < 1, i.e., if ln x, or x 91, a < a. Similarly, () ( ) {( )} for the denominator to be non-zero and positive, it is necessary that ln x d 1, i.e., ln x 2, or x 8. Again, numerator denominator d (1 d), lnx > 0 as d < 1. Hence for x 8, a >a. We have thus shown that for x 91 and n [ln x] 1, a < a, a >a. Thus either a or a is the largest term, i.e., a attains the maximum value at n [ln x] 1 or at n [ln x] 2 for x 91. Generally, it is at n [ln x] 1 that x10 2

3 attains the maximum value but occasionally as is the case for m3 and m10, it may be at n [ln x] 2. It is thus clear that the terms in (5) first increase, attain the maximum value and then begin decreasing to approach zero. Theorem 2: a ()! attains the minimum value at [ln x] 1, where [ln x] denotes the greatest integer contained in ln x. Proof: ;. Putting n[ln x] 1, [ ] 1 ; [ ] <1. i.e.,a a < a, i.e., a attains the least possible value at n [ln x] 1. This theorem thus shows that the terms in (6) first decrease, attain the minimum value and then start increasing to approach. According to (4), Li(x) ln ln t 1 x 2 i.e., Li(x) ln ln x ln ln 2 (7) < ln 2 We can see that ln 2 ln 2 /(1- ) Actually, ; ln ln Consequently, the second (-) quantity comes to (correct to 6 decimal places!). The third quantity (.468) may be ignored by leaving out infinitely many smaller terms (whose sum approximately equals this quantity) of the second quantity. If Li(x) ln ln x, then denoting the remainder by R(x), we have R(x) < () ().()! ().()! ). ().()! ().()! 1 () () If k > ln x, then R(x) ().()! 1/ (1 () ) ().()! {(k 1) ln x }. 3

4 So if k e ln x 1, R(x) <.633. Hence both quantities then cancel almost out each other (with error <.165). Hence I take k [2.72 (ln x)] 1 which gives the approximation formula: [2.72(ln x)]1 Li(x) ln ln x lnn x 1 (7a) n.n! Now according to (5), Li(x) x dt 2 ()! x 1 2 i.,e., Li(x) ( ()! ) ( ()! ) (8) Both quantities tend to here. The terms in first series initially decrease before they start increasing; the second is (strictly) monotonically increasing series. I have found that, except for some smaller values of x(<1000), (8) can be truncated, without any significant loss in accuracy, to this approximation formula: [ln x]1 (n1)!x Li(x) n1 (8a) ln n x Though the formula (7a) is slightly more accurate than (8a) for smaller values of x (< 1000), it requires roughly (2.7) times the number of terms needed in the latter, which is more convenient for larger values of x. I give an illustrative example here: Let x10. According to formula (7a): Li(10 [.( ) ln lnx )] Here 20 terms were needed for the computation. Since ln(10 ) 6.9, [ln x] 1 7. Thus only 7 terms would be required for computation with formula (8a). ()! Li (10 ) The following table contain certain values of Li(x) computed by me with the help of formulae (7a) and (8a). These values compare very well with the values of Li(x) borrowed from various sources [2] to [7] and thereby validate my formulae. 4

5 Table of values [2.72(ln x)]1 x Li(x) ln ln x lnn x [ln x]1 (n1)!x 1 n.n! n1 ln n x ? Note: My calculations were done on Microsoft Excel for x< 10. But the Program gives only rounded figures (inexact calculations) for x 10. So I had to use Scientific Calculator manually for x 10. References: 1. Apostol, Tom M. Introduction to Analytic Number Theory, Springer International Student Edition, Narosa Publishing House, New Delhi, 1995 reprint, Exercises for Chapter4,19, pp Burton, David M. Elementary Number Theory, Tata Mcraw-Hill,2007, New Delhi, 6 th ed., p.374 & Caldwell, Chris K. Prime Pages, 4. Richard Crandall & Carl Pomerance, Prime Numbers: A Computational Perspective, 2 nd ed., Springer (2005), New York, pp Hua Loo Keng, Introduction to Number Theory, Springer-Verlag, Berlin, 1982, p Ore, Oystein, Number Theory and Its History, McGraw Hill Book Company, New York, 1948, p USpensky, J.V. & Heaselet, M.A. Elementary Number Theory, McGraw Hill Book Company, New York, 1939, pp ****** 5