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

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

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

B

_{0}=1 ; B_{1}= -1/2 ; B_{2}= 1/6 ....B

_{s}= 0 for s ≥ 3 and oddBernoulli numbers can be calculated recursivly by solving :

for B

_{n}.

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

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

For even

sI used the direct formula :in which

Bare the Bernoulli numbers._{s}For odd values of

swe 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

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, 2001Sebah 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≥2is 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

c

_{k}for k=2 to k=1000 with 110 digits accuracy, my own calculationsThe Hardy-Littlewood constants were evaluated by splitting the product term into two :

Here

Mis a suitably chosen parameter,M=8 kserved 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)which for^{s}p > 8 kis sufficiently fast. Instead of direct evaluation ( and truncation ) we use :where

ζis the Riemann zeta prime function._{P}(s)

- Data at other websites :
- c
_{k}k=2 to k=16 : Robert Joseph Harley

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 < m_{1}< m_{2}.... < m_{k-1}< rr = half-gap size

wis the number of distinct residues for a single permutation of the set of integers {_{m,r}(q)0 m} modulo the odd prime_{1}m_{2}.... m_{k-1}rq.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-ksmaller ifr+1is 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 althoughT(r,k) = 0for k>N with N depending onr, see table for T(r,N) and N and cpu-time as function ofr.

- Data at other websites :
- T(r,k) for r=1 through 70 : Robert Harley

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, Hardy-Littlewood constants, see above section (2.8)_{k}

T(r,k), see above section (2.9)

For both equations the product goes over the odd primes

qin 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=1is of importance. ForT(r,k)there are nomto permute and_{k}Zwhich when combined with_{1}(1) = 1Zgives the simple equation :_{2}(1)

Here c

_{2}=0.6601618158468695739278 .... is the well-known twin-prime constant. The product term in this equation extends only over those odd primes less or equal torwhich are factors of the half-gap sizerand that it is equal to 1 ifris equal to a power of 2.

for r ≤ 128r 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

- 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 :

H_{n}= 1 + 1/2 + 1/3 + 1/4 ... + 1/n

to be continued .....

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=10

^{500}: Li=0.86934.. * 10^{497}while x/ln(x) = 0.86859... * 10^{497}## Recursion relationships for higher logarithmic integrals :

Repeated application :

The following equation was in principle first derived by Richard Brent in :

We define :

"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 LittlewoodL(r,x)= number of conjectured occurrences of gaps of size2 rbetween successive primes &lex.

For the definition of function

If one follows the ideas set forth by Hardy and Littlewood the lower integration limit would besee section (2.10) above, the integration of the natural logarithm to the power (k+1) follows the equations outlined in section (2.12).A(r,k)a= 2. This value is fine for small values of the half-gap size but fails for higher values ofr. To be continued.

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