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

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

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

2.4 Bernoulli Numbers

B0=1 ; B1 = -1/2 ; B2 = 1/6 ....

Bs = 0 for s ≥ 3 and odd

Bernoulli numbers can be calculated recursivly by solving :

for Bn.

For s = 1 to s = 500


2.5 Moebius Function

Defined as :

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

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 Bs 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

2.7 Riemann zeta prime function summed over all primes

ζP[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

2.8 Hardy-Littlewood constants

ck 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 ζP(s) is the Riemann zeta prime function.

Data at other websites :
ck k=2 to k=16 : Robert Joseph Harley

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 mi, 1 ≤ i ≤ k-1, subject to the condition that :

0 < m1 < m2 .... < mk-1 < r

r = half-gap size

wm,r(q) is the number of distinct residues for a single permutation of the set of integers {0 m1 m2 .... mk-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=95 (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

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 :
ck , 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 mk 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
rComposition of r
48/15 = 3.200105r=3*5*7
8/3 = 2.66715 , 30 , 45 , 60 , 75 , 90 r=2^i * 3^j * 5^m ; i≥0 j,m≥1
12/5 = 2.40021 , 42 , 63 , 84 , 126 r=2^i * 3^j * 7^m ; i≥0 j,m≥1
20/9 = 2.22233 , 63 , 99 r=2^i * 3^j * 11^m ; i≥0 j,m≥1
24/11 = 2.181839 , 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=1.008127r=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


2.11 γ ≈ 0.57721 56649 01532 ... , Euler-Mascheroni gamma constant

Data at other websites :
= 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 : wikipedia
http://pi.lacim.uqam.ca/ Lists γ² , γ³ , and exp(γ)
http://numbers.computation.free.fr provides several methods to determine the value of γ , not the worst of it is based on the harmonic number notation :
Hn = 1 + 1/2 + 1/3 + 1/4 ... + 1/n
to be continued .....

2.12 Off-set logarithmic Integral      

We evaluate li(x) using a formula due to Ramanujan :


which is convergent for all values of x.
See wikipedia and Wolfram.com
In above equation ⌊x⌋ is the floor function ( its value is the largest integer ≤ x ).
Note that, if one is only interested in calculating Li(x) using the function li(x), the value of the Euler-Mascheroni gamma constant is irrelevant.

In the limit of x going to infinity : Li(x) -> x/ln(x) but the convergence is slow. At x=10500 : Li=0.86934.. * 10497 while x/ln(x) = 0.86859... * 10497

Recursion relationships for higher logarithmic integrals :


Repeated application :


2.13 Conjectured Frequency of Occurrence of Prime Gaps

The following equation was in principle first derived by Richard Brent in :
"The Distribution of Small Gaps Between Successive Primes",
Mathematics of Computation, volume 28, number 125, January 1974.

His work was based on the k-tuple conjecture by Hardy and Littlewood

We define : L(r,x) = number of conjectured occurrences of gaps of size 2 r between successive primes &le x.

For the definition of function A(r,k) see section (2.10) above, the integration of the natural logarithm to the power (k+1) follows the equations outlined in section (2.12).

If one follows the ideas set forth by Hardy and Littlewood the lower integration limit would be a= 2. This value is fine for small values of the half-gap size but fails for higher values of r. To be continued.

Zig Herzog; hgn@psu.edu
Last revised: 11/21/08