“Prime Counting Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Prime Counting Function

释义

Prime Counting Function

The function giving the number of Primes less than (Shanks 1993, p. 15). The first few values are 0, 1, 2, 2,3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ... (Sloane's A000720). The following table gives the values of for powers of 10(Sloane's A006880; Hardy and Wright 1979, p. 4; Shanks 1993, pp. 242-243; Ribenboim 1996, p. 237). Deleglise and Rivat (1996) havecomputed .

is incorrectly given as 50,847,478 in Hardy and Wright (1979). The prime counting function can be expressed byLegendre's Formula, Lehmer's Formula, Mapes' Method, or Meissel's Formula. A brief history ofattempts to calculate

is given by Berndt (1994). The following table is taken from Riesel (1994).

Method	Time	Storage
Legendre
Meissel
Lehmer
Mapes'
Lagarias-Miller-Odlyzko
Lagarias-Odlyzko 1
Lagarias-Odlyzko 2

A modified version of the prime counting function is given by

where

is the Möbius Function and

is the Riemann-Mangoldt Function.

The notation is also used to denote the number of Primes of the form (Shanks 1993, pp. 21-22).Groups of Equinumerous values of include (, ), (, ),(, , , ), (, ), (, , , , , ), (, , , ), (, , , , , ), and so on. The values of for small are given in the following table for the first fewpowers of ten (Shanks 1993).


1	2	1	2
11	13	11	13
80	87	80	87
611	617	609	619
4784	4807	4783	4808
39231	39266	39175	39322
332194	332384	332180	332398


0	2	1	0
5	7	7	5
40	47	42	38
306	309	310	303
2387	2412	2402	2390
19617	19622	19665	19593
166104	166212	166230	166032


	1	1
	11	12
	80	86
	611	616
	4784	4806
	39231	39265


0	1	1	0	1	0
3	4	5	3	5	4
28	27	30	26	29	27
203	203	209	202	211	200
1593	1584	1613	1601	1604	1596
13063	13065	13105	13069	13105	13090


0	1	1	1
5	7	6	6
37	44	43	43
295	311	314	308
2384	2409	2399	2399
19552	19653	19623	19669
165976	166161	166204	166237

Note that since , , , and are Equinumerous,

are also equinumerous.

Erdös proved that there exist at least one Prime of the form and at least one Prime of theform between and for all .

The smallest such that for , 3, ... are 2, 27, 96, 330, 1008, ... (Sloane's A038625), and thecorresponding are 1, 9, 24, 66, 168, 437, ... (Sloane's A038626). The number of solutions of for, 3, ... are 4, 3, 3, 6, 7, 6, ... (Sloane's A038627).

See also Bertelsen's Number, Equinumerous, Prime Arithmetic Progression, Prime Number Theorem, Riemann Weighted Prime-Power Counting Function

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 134-135, 1994.

Brent, R. P. ``Irregularities in the Distribution of Primes and Twin Primes.'' Math. Comput. 29, 43-56, 1975.

Deleglise, M. and Rivat, J. ``Computing : The Meissel, Lehmer, Lagarias, Miller, Odlyzko Method.'' Math. Comput. 65, 235-245, 1996.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html

Forbes, T. ``Prime -tuplets.'' http://www.ltkz.demon.co.uk/ktuplets.htm.

Guiasu, S. ``Is There Any Regularity in the Distribution of Prime Numbers at the Beginning of the Sequence of Positive Integers?'' Math. Mag. 68, 110-121, 1995.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Lagarias, J.; Miller, V. S.; and Odlyzko, A. ``Computing : The Meissel-Lehmer Method.'' Math. Comput. 44, 537-560, 1985.

Lagarias, J. and Odlyzko, A. ``Computing : An Analytic Method.'' J. Algorithms 8, 173-191, 1987.

Mapes, D. C. ``Fast Method for Computing the Number of Primes Less than a Given Limit.'' Math. Comput. 17, 179-185, 1963.

Meissel, E. D. F. ``Über die Bestimmung der Primzahlmenge innerhalb gegebener Grenzen.'' Math. Ann. 2, 636-642, 1870.

Ribenboim, P. The New Book of Prime Number Records, 3rd ed. New York: Springer-Verlag, 1996.

Riesel, H. ``The Number of Primes Below .'' Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 10-12, 1994.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.

Sloane, N. J. A. Sequences A038625,A038626,A038627,A000720/M0256, andA006880/M3608in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 74-76, 1991.

Wolf, M. ``Unexpected Regularities in the Distribution of Prime Numbers.'' http://rose.ift.uni.wroc.pl/~mwolf/.

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