I'll try my best to reconstruct how this appears in my pdf:
$$d:denom(B_n)=\prod_{p-1|n} p$$
the "$p-1|n$" part, I don't understand. I get this has something to do with Riemann Zeta Function, and prime numbers, but I'm new to capital pi notation, and this is confusing.
I am to understand this is a component for calculating the denominator in Bernoulli numbers. The $n$ is the same as the $n$ in:
$$B_n$$
for even Bernoulli numbers.
The "$|n$" part is the most confusing, and I don't know how to start "$p$", but presume it implies a prime number?
If $n$ is $50$, I'm supposed to get "divisors of $n$ are $1$, $2$, $5$, $10$, $25$, $50$, and hence $d=(2)(3)(11) = 66$".
But that doesn't make any sense to me where the $1-50$ came from (they are factors of $50$) or why he chose the prime numbers $2$, $3$, and $11$, but not $7$? I do understand capital pi involves multiplying, but I don't know why he stops at $11$ or why he skips $7$, or what that has to do with the factors of $50$.
(reference: https://wstein.org/edu/fall05/168/projects/kevin_mcgown/bernproj.pdf)
Best Answer
The notation $a|b\;$means $b$ is a multiple of $a$, or equivalently, $a$ is a divisor of $b$.
It's read as "$a$ divides $b$".
Thus, the product in question is the product of all primes $p$ such that $p-1$ divides $n$.
For example, \begin{align*} \prod_{p-1|12} p&=(2)(3)(5)(7)(13)=2730\\[4pt] \prod_{p-1|20} p&=(2)(3)(5)(11)=330\\[4pt] \end{align*} For the example you mentioned, $n=50$, the divisors of $50$ are, as you noted, $1,2,5,10,25,50$, but of those, only $1,2,10$ are the form $p-1$ for some prime $p$ (since those are the only ones from that list which are one less than a prime), hence $$\prod_{p-1|50} p=(2)(3)(11)=66$$
As regards the capital "Pi" notation, it's the product, starting with a default initial value of $1$, of the values of the expression to the right of the product symbol, where the expression is evaluated and used as a factor for each case of the specified iteration condition (i.e., you get a factor for each case where the iteration condition is satisfied).
For the product in question, for a given value of $n$, the specified condition is the requirement that $p-1$ divides $n$, together with the assumption that in this context, $p$ is also required to be prime. For each $p$ satisfying the condition, you get a factor of $p$ (since the expression to the right of the product symbol is $p$).