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The Prime Counting Function and The nth Term of Prime Number Series
H.V. Kumarkhaniya1
Section:Research Paper, Product Type: Journal-Paper
Vol.7 ,
Issue.3 , pp.82-88, Jun-2020
Online published on Jun 30, 2020
Copyright © H.V. Kumarkhaniya . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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IEEE Style Citation: H.V. Kumarkhaniya, “The Prime Counting Function and The nth Term of Prime Number Series,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.82-88, 2020.
MLA Style Citation: H.V. Kumarkhaniya "The Prime Counting Function and The nth Term of Prime Number Series." International Journal of Scientific Research in Mathematical and Statistical Sciences 7.3 (2020): 82-88.
APA Style Citation: H.V. Kumarkhaniya, (2020). The Prime Counting Function and The nth Term of Prime Number Series. International Journal of Scientific Research in Mathematical and Statistical Sciences, 7(3), 82-88.
BibTex Style Citation:
@article{Kumarkhaniya_2020,
author = {H.V. Kumarkhaniya},
title = {The Prime Counting Function and The nth Term of Prime Number Series},
journal = {International Journal of Scientific Research in Mathematical and Statistical Sciences},
issue_date = {6 2020},
volume = {7},
Issue = {3},
month = {6},
year = {2020},
issn = {2347-2693},
pages = {82-88},
url = {https://www.isroset.org/journal/IJSRMSS/full_paper_view.php?paper_id=1939},
publisher = {IJCSE, Indore, INDIA},
}
RIS Style Citation:
TY - JOUR
UR - https://www.isroset.org/journal/IJSRMSS/full_paper_view.php?paper_id=1939
TI - The Prime Counting Function and The nth Term of Prime Number Series
T2 - International Journal of Scientific Research in Mathematical and Statistical Sciences
AU - H.V. Kumarkhaniya
PY - 2020
DA - 2020/06/30
PB - IJCSE, Indore, INDIA
SP - 82-88
IS - 3
VL - 7
SN - 2347-2693
ER -
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Abstract :
earliest works on the prime numbers are as old as 1500 BC yet as of the present, no known simple formula can distinguish the prime numbers from the composite numbers. The number of primes not greater than any natural number z is given by a function denoted by π(z), known as prime counting function. The value of which is approximated using the prime number theorem. It can be also computed using a function based on the work of B. Riemann. This article includes, derivation of simple formula of function that counts prime upto any given number z with essential calculations and an example to understand. The presented theoretical work is also aimed to derive the general formula for the nth term of the prime number series which is an exact formula for the value of nth prime. The nth prime can be computed using the derived formula and the algorithm presented in this article.
Key-Words / Index Term :
prime numbers, prime counting function, nth prime number, composite numbers, number theory
References :
[1] J. Williamson (translator and commentator), ?The Elements of Euclid,? Clarendon Press, Oxford, pp. 63, 1782.
[2] R.S. Horsley, ?The Sieve of Eratosthenes. Being an account of his method of finding all the prime numbers,? Philosophical Transactions, Vol.62, pp. 327-347, 1772.
[3] C.F. Gauss, ?K?nigliche Gesellschaft der Wissenschaften zu G?ttingen,? Werke, Vol.2, pp. 444-447, 1863.
[4] A.M. Legendre, ?Essai sur la th?orie des nombres,? 2nd edition, Courcier, Paris, pp. 394, 1808.
[5] B. Riemann, ??ber die Anzahl der Primzahlen unter einer gegebenen Gr?sse,? Monatsberichte der Berliner Akademie, pp. 671-680, Nov. 1859.
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