Mathematical Constants and Functions related to Prime Gap Conjecture

2.1 e = 2.718....

First 5000 digits, my own calculations

Formula evaluated : e = 1+1/1!+1/2!+...+1/k!+...

Data at other websites :

http://bootes.math.uqam.ca/piDATA/exp1.txt

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/e_10000.html

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2.2 pi = 3.1415....

First 5000 digits, my own calculations

Formula evaluated : Pi/4 = 4*arctan(1/5)-arctan(1/239) and
arctan(x) = x - (1/3)*x^3 + (1/5)*x^5 - (1/7)*x^7 ....

Data at other websites :

http://bootes.math.uqam.ca/piDATA/pi10000.txt

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2.3 e^pi = 23.14069....

First 5000 digits, my own calculations

Formula evaluated : e^pi = 1+pi/1!+pi^2/2!+...+pi^k/k!+..

Data at other websites :

http://pi.lacim.uqam.ca/piDATA/exppipi.txt

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2.4 Bernoulli Numbers

B₀=1 ; B₁ = -1/2 ; B₂ = 1/6 ....

B_s = 0 for s ≥ 3 and odd

Bernoulli numbers can be calculated recursivly by solving :

for B_n.

For s = 1 to s = 500

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2.5 Moebius Function

Defined as :

μ[n]	=	+1	if n = 1
	=	0	if n contains at least one squared factor
	=	+1	if n is the product of an even number of distinct primes
	=	-1	if n is the product of an odd number of distinct primes

For n = 1 to n = 10000, my own calculations

Data at other websites :

http://www.research.att.com/~njas/sequences/b008683.txt

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2.6 Riemann zeta function summed over all integers and for s ≥ 2

ζ[s] for s=2 to s=3200 with 3200 digits accuracy, my own calculations (10 MBytes)

For even s I used the direct formula :

in which B_s are the Bernoulli numbers.

For odd values of s we used two rapidly converging series given by :
Henri Cohen : High-Precision Computation of Hardy-Littewood constants ( Look on his page for above title, the respective file is in .dvi format as of Aug. 2007 ).

Data at other websites :

ζ[2] : http://pi.lacim.uqam.ca/piDATA/zeta2.txt
ζ[4] : http://pi.lacim.uqam.ca/piDATA/zeta4.txt
ζ[s] s=3,5,7... 99 : http://pi.lacim.uqam.ca/piDATA/zetaflat.txt

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2.7 Riemann zeta prime function summed over all primes

ζ_P[s] for s=2 to s=500 with 200 digits accuracy, my own calculations (100 kBytes)

For equations used I refer to :
Pascal Sebah and Xavier Gourdon : Some Constants from Number Theory; Nov. 27, 2001

Sebah and Gourdon propose :

in which μ(n) is the Moebius function and ζ(M,t) is given by :

ζ(t) denotes the Riemann zeta function and M≥2 is an integer value which increases the rate of convergence as it gets bigger.

Data at other websites :

Pascal Sebah and Xavier Gourdon : Some Constants from Number Theory; Nov. 27, 2001 for s = 2 through 8

http://www.research.att.com/~njas/sequences Search for "Riemann zeta prime" on this page

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2.8 Hardy-Littlewood constants

c_k for k=2 to k=1000 with 110 digits accuracy, my own calculations

The Hardy-Littlewood constants were evaluated by splitting the product term into two :

Here M is a suitably chosen parameter, M=8 k served us well. The first product term is evaluated directly using only a finite number of primes which can be done to any desired accuracy using a multi-precision math library. The second product term :

can be evaluated using :

This series converges like (k/p)^s which for p > 8 k is sufficiently fast. Instead of direct evaluation ( and truncation ) we use :

where ζ_P(s) is the Riemann zeta prime function.

Data at other websites :

c_k k=2 to k=16 : Robert Joseph Harley

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2.9 T(r,k) function

First introduced by Richard Brent in :
"The Distribution of Small Gaps Between Successive Primes",
Mathematics of Computation, volume 28, number 125, January 1974.

Defined as :

In above equation the summation stretches over all possible permutations of the integers m_i, 1 ≤ i ≤ k-1, subject to the condition that :

0 < m₁ < m₂ .... < m_k-1 < r

r = half-gap size

w_m,r(q) is the number of distinct residues for a single permutation of the set of integers {0 m₁ m₂ .... m_k-1 r} modulo the odd prime q.

Note that my definition of T(r,k) is slightly different from that given by Brent and calculated by Harley. As a result my values are by a factor of r-k smaller if r+1 is prime.

For r=1 up to r=92 (and counting); my own calculations and those by Tomas Oliveira e Silva

Computationally, the function T(r,k) is most time-consuming to evaluate although T(r,k) = 0 for k>N with N depending on r, see table for T(r,N) and N and cpu-time as function of r.

Data at other websites :

T(r,k) for r=1 through 70 : Robert Harley

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2.10 A(r,k) function

First introduced by Richard Brent in :
"The Distribution of Small Gaps Between Successive Primes",
Mathematics of Computation, volume 28, number 125, January 1974.

Defined as :

with :
c_k , Hardy-Littlewood constants, see above section (2.8)
T(r,k) , see above section (2.9)

For both equations the product goes over the odd primes q in the indicated ranges.
Computationally, the function T(r,k) is most time-consuming to evaluate and restricts the availability of values for the function A(r,k).

The special case of k=1 is of importance. For T(r,k) there are no m_k to permute and Z₁(1) = 1 which when combined with Z₂(1) gives the simple equation :

Here c₂=0.6601618158468695739278 .... is the well-known twin-prime constant. The product term in this equation extends only over those odd primes less or equal to r which are factors of the half-gap size r and that it is equal to 1 if r is equal to a power of 2.

    for r ≤ 128

	r	Composition of r
48/15 = 3.200	105	r=3*5*7
8/3 = 2.667	15 , 30 , 45 , 60 , 75 , 90	r=2^i * 3^j * 5^m ; i≥0 j,m≥1
12/5 = 2.400	21 , 42 , 63 , 84 , 126	r=2^i * 3^j * 7^m ; i≥0 j,m≥1
20/9 = 2.222	33 , 63 , 99	r=2^i * 3^j * 11^m ; i≥0 j,m≥1
24/11 = 2.1818	39 , 78 , 117	r=2^i * 3^j * 13^m ; i≥0 j,m≥1
⋅	⋅	⋅
2/1 = 2	3 , 6 , 9 , 12 , 18 , 24 , 27 , 36 , 48 , 54 , 72 , 81 , 96 , 108	r=2^i * 3^j ; i≥0 j≥1
⋅	⋅	⋅
126/125=1.008	127	r=127
1	1 , 2 , 4, 8 , 16 , 32 , 64 , 128	r = 2^i ; i≥0

Known values of A(r,k) (in progress); my own calculations

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