How do you compute the singular series?

Question

Terence Tao gives at his blog the following formula for something called the singular series:

$$\large\mathfrak{S}(h)=2\Pi_{2}\prod\limits_{p|h;p>2}\frac{p-2}{p-1}$$

where $\Pi_{2}=0.66016...$ is the twin prime constant.

I can't understand how to compute the product: $$\prod\limits_{p|h;p>2}\frac{p-2}{p-1}$$

Could you possibly compute numerically a few examples and maybe explain what $$p|h;p>2$$ means?

Usually $p|h$ means $p$ divides $h$, but since $h$ can be a small integer, I don't understand how it works.

Answer 1

I believe it just means to use all of the odd prime numbers, $p$, which are factors of $h$ in the product of $\frac{p-2}{p-1}$. Thus, for example, if $h = 90 = 2 \times 3^2 \times 5$, then the odd prime factors $p$ are $3$ and $5$, thus giving that

$$\prod\limits_{p|h;p>2}\frac{p-2}{p-1} = \left(\frac{1}{2}\right)\left(\frac{3}{4}\right) = \frac{3}{8} \tag{1}\label{eq1}$$

Answer 2

John Omielan's answer as a Mathematica program would be:

Clear[a, b, n, j, i]
Table[Times @@ (DeleteDuplicates[
      DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 
       2]] - 2)/
  Times @@ (DeleteDuplicates[
      DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 
       2]] - 1), {h, 1, 100}]
Numerator[%]
aa = Denominator[%%]

The output as fractions:
{1, 1, 1/2, 1, 3/4, 1/2, 5/6, 1, 1/2, 3/4, 9/10, 1/2, 11/12, 5/6, \
3/8, 1, 15/16, 1/2, 17/18, 3/4, 5/12, 9/10, 21/22, 1/2, 3/4, 11/12, \
1/2, 5/6, 27/28, 3/8, 29/30, 1, 9/20, 15/16, 5/8, 1/2, 35/36, 17/18, \
11/24, 3/4, 39/40, 5/12, 41/42, 9/10, 3/8, 21/22, 45/46, 1/2, 5/6, \
3/4, 15/32, 11/12, 51/52, 1/2, 27/40, 5/6, 17/36, 27/28, 57/58, 3/8, \
59/60, 29/30, 5/12, 1, 11/16, 9/20, 65/66, 15/16, 21/44, 5/8, 69/70, \
1/2, 71/72, 35/36, 3/8, 17/18, 3/4, 11/24, 77/78, 3/4, 1/2, 39/40, \
81/82, 5/12, 45/64, 41/42, 27/56, 9/10, 87/88, 3/8, 55/72, 21/22, \
29/60, 45/46, 17/24, 1/2, 95/96, 5/6, 9/20, 3/4}

of which the numerators are:

{1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, \
21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 5, 1, 35, 17, 11, 3, 39, 5, \
41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, \
29, 5, 1, 11, 9, 65, 15, 21, 5, 69, 1, 71, 35, 3, 17, 3, 11, 77, 3, \
1, 39, 81, 5, 45, 41, 27, 9, 87, 3, 55, 21, 29, 45, 17, 1, 95, 5, 9,
  3}

and the denominators are:
{1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, \
10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 1, 20, 16, 8, 2, 36, 18, 24, 4, \
40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, \
8, 60, 30, 12, 1, 16, 20, 66, 16, 44, 8, 70, 2, 72, 36, 8, 18, 4, 24, \
78, 4, 2, 40, 82, 12, 64, 42, 56, 10, 88, 8, 72, 22, 60, 46, 24, 2, \
96, 6, 20, 4}