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.
1. 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)! method.
2. lim dn/pn = 0
3. 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 ).
Examples from numerical calculations :
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
1. Gerhard Niklasch and Pieter Moree 1000-digit values for c2,c3,c4, and c5, Sept 1999. My own copy
2. Robert Harley
3. The H-L constants c_2 through c_16, 45 digit values. 1994 My own copy
4. Steven Finsch Currently ( Jan 2003 ) un available but praised by many.
5. "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
1. 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).
2. Best values in the literature ( 26 digits ), see Abramovitz & Stegun , p 811)
Computation of pi = 3.1415...
1. The 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.
2. 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...
1. The first 2 million digits, NASA related site
2. Some programs to determine e ( I have a copy of eclassic.c )

### Presentations, Publications, my own

Zig Herzog; hgn@psu.edu
Last revised: 03/01/01