The Prime Counting Function

Sang-Un, Lee; MyeongBok Choi

Journal of The Korea Society of Computer and Information

Abbr : JKSCI
2011, 16(10), pp.101-110
Publisher : The Korean Society Of Computer And Information
Research Area : Computer Science

Sang-Un, Lee ¹ , MyeongBok Choi ²

¹ 강릉원주대학교
² 강릉원주대학교

ABSTRACT

The Riemann's zeta function has been known as answer for a number of primes less than given number . In prime number theorem, there are another approximation function , and . The error about is . The logarithmic integral function is . This paper shows that the can be represent with finite , and presents generalized prime counting function . Firstly, the can be represent to and such that . Then, is adjusted by with and error compensation value . As a results, this paper get the for . Then, this paper suggests a generalized function . The function superior than Riemann's zeta function in representation of prime counting.
KEYWORDS

prime number theorem , Riemann zeta function , natural logarithmic function , offset logarithmic integral function
REFERENCES (8)

[journal] D. Zagier / 1997 / Newman's Short Proof of the Prime Number Theorem / American Mathematical Monthly 104 (8) : 705~708

[journal] A. O. L. Atkin / 2004 / Prime Sieves Using Binary Quadratic Forms / Mathematics of Computation 73 : 1023~1030

[book] B. Riemann / 1998 / Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, in On the Number of Primes Less Than a Given Quantity / Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin

[journal] J. M. Borwein / 2000 / Computational Strategies for the Riemann Zata Function / Journal of Computational Applied Mathematics 121 : 247~296

[journal] T. Kotnik / 2008 / The Prime-counting Function and its Analytic Approximations / Advanced Computat ional Mathematics 29 (1) : 55~70

[journal] D. Goldfeld / 2009 / The Elementary Proof of the Prime Number Theorem: An Historical Perspective / The Mathematical Intelligencer 31 (3) : 18~23

[book] N. M. Temme / 2010 / Exponential, Logarithmic, Sine, and Cosign Integrals / Cambridge University Press

[book] G. H. Hardy / 1979 / An Introduction to the Theory of Numbers / Oxford University Press : 355~356
KCI Citation (0)

KJC Korea
Journal Central

Statistics

Tools