Distribution of Prime Gaps
Mathematical constants and functions related to prime-gap conjecture (under development)
Distribution of primes numbers
- Fundamental theorem
- There are infinitely many prime numbers
- Prime counting Function
- Let pi(x) denote the function indicating the number of primes p
such that :
p <= x
An approximation due to Gauss (1792) is :
pi(x) ~ int (t=2 to t=x) dt/ln(t) = logarithmic integral ln= natural log
The largest computed value of pi(x) (so the exact value) is at x=4*10^16
(1985 Lagarias, Miller & Odlyzko , Computing pi(x) : The Meissel-Lehmer method.
Math. Comp.,44,1985,537-560). There are 1,075,292,778,753,150 prime numbers
below the value of x=4*10^16.
A nice entry page
to references concerning computing pi(x) is given by
Chris K. Caldwell.
I am trying to my own summary together
focusing on the range up to 10^17
- Gaps between prime numbers
- Differentiating pi(x) gives us a first approximation for the
Average gap between adjacent prime numbers = ln(x)
Let dn = pn+1 - pn the difference between
adjacent prime numbers. Some estimates are known concerning dn
as n goes to infinity.
Examples from numerical calculations :
- lim sup dn = infinity ( easy proof : for every number N you
can construct N consecutive non-prime numbers. They are
(N+1)! + 2; (N+1)! + 3; ...... ; (N+1)! + (N+1)
Example : N=5 : 722, 723, 724, 725, 726 are 5 consecutive non-prime
number. Note though that 24, 25, 26, 27, 28 are also 5 consecutive
non-primes but at much lower value than indicated by the (N+1)!
- lim dn/pn = 0
- Presently research (not mine) is under way to show that
dn = O(pnTheta)
The lowest value assured for Theta is 85/164 (Lou & Yao, 1985 :
The upper bound of the difference between consecutive primes,
Kexue Tongbao, 8, 1985, 128-129 )
Theta = 1051/1920 was obtained by Mozzochi, 1986 (Mozzochi,C.J., On the
difference between consecutive primes. J.Nb. Th.,24,1986,181-187 ).
d = 804 between 90,874,329,412,297 and the next prime number ( Young & Potler
, First occurrence of prime gaps, Math. Comp.,55,1989,383-389 )
- Hardy-Littlewood constants
- Gerhard Niklasch and Pieter Moree 1000-digit values for
c2,c3,c4, and c5, Sept 1999.
My own copy
- Robert Harley
The H-L constants c_2 through c_16, 45 digit values. 1994
My own copy
- Steven Finsch Currently ( Jan 2003 ) un available but praised by many.
- "High-Precision Computation of Hardy-Littlewood Constants , by Henry Cohen"
Have the hard copy of his pre-printi and on disk www/primes/hardylw.dvi hardylw.ps.
- Riemann Zeta Function
- A good resource by the creator of Mathematica :http://mathworld.wolfram.com with fast
converging series expansion of odd values of argument ( equation 75 and 76).
- Best values in the literature ( 26 digits ), see Abramovitz & Stegun ,
- Computation of pi = 3.1415...
first 50,000 digits of pi by Roy Williams, CalTech.
My own copy I also have a C-program
by the same author ( he called it pi.c , I renamed it 3.14.c which calculates
pi to any number of desired digits.
- 3.14 to 25000 digits by J.Borwein for the
Centre for Experimental and Constructive Mathematics, Departments
of Mathematics and
Statistics, Simon Fraser University
- Computations on e = 2.718281828...
- The first 2 million digits, NASA related site
- Some programs to determine e ( I have a copy of eclassic.c )
Data obtained with package lowp
- Maximal Prime Gap as function
of prime number.
- First Occurrences of Prime Gaps
as functions of gap size and preceeding prime number.
- Prime Count pi(x)
- Counting functions for various prime gaps, tables
Gap : 2 ,
more to come ...
- Counting functions for various prime gaps,
gaps 2,4,6,8,10 , graph for 10,20,40,60,80,100
Manual for lowp package : Determine primes numbers
- Overview of programs in this package
- ini file for program lowp
- Structures of files containing primes
- The pre-sieve method for storing primes
- Tests run on lowp to generate primes
Manual for mpz = multi-precision package
- Overview of current version : 1.0
- Conversion Functions
- Misc. : printing, comparing of mpz-numbers and more
Various proofs on primality as found on the web
- Simpler primality test for Sophie-Germain numbers
- Generalization of Euler -Lagrange theorem and new
Presentations, Publications, my own
- Faculty Colloquium, April 2002 , Mont Alto
Zig Herzog; email@example.com
Last revised: 03/01/01