Mathematical constants and functions related to prime-gap conjecture (under development)

**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 d_{n}= p_{n+1}- p_{n}the difference between adjacent prime numbers. Some estimates are known concerning d_{n}as n goes to infinity.

- lim sup d
_{n}= 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. - lim d
_{n}/p_{n}= 0 - Presently research (not mine) is under way to show that

d_{n}= O(p_{n}^{Theta})

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 )

- lim sup d
**Hardy-Littlewood constants**- Gerhard Niklasch and Pieter Moree 1000-digit values for
c
_{2},c_{3},c_{4}, and c_{5}, 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.

- Gerhard Niklasch and Pieter Moree 1000-digit values for
c
**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 , p 811)

**Computation of pi = 3.1415...**-
- 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.
- 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 )

- 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 , 4 , 6 , 8 , 10 , more to come ... - Counting functions for various prime gaps,

graph for gaps 2,4,6,8,10 , graph for 10,20,40,60,80,100

- 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

- Overview of current version : 1.0
- Addition/Subtraction
- Multiplication
- Division
- Conversion Functions
- Misc. : printing, comparing of mpz-numbers and more

- Simpler primality test for Sophie-Germain numbers
- Generalization of Euler -Lagrange theorem and new primality tests

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