Definition:Ramanujan Phi Function

Definition

The two argument Ramanujan phi function, $\map \phi {a, n}$, is defined for $a \in \Z, a > 1$ as:

$\ds \map \phi {a, n} = 1 + 2 \sum_{k \mathop = 1}^n \dfrac 1 {\paren {a k }^3 - a k}$

Ramanujan Phi Function One Argument

When $n$ is omitted, the one argument Ramanujan phi function, $\map \phi a$, is defined as:

\(\ds \map \phi a\)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \map \phi {a, n}\)

\(\ds \)

\(=\)

\(\ds 1 + 2 \sum_{k \mathop = 1}^{\infty} \dfrac 1 {\paren {a k }^3 - a k}\)

Also see

• Results about the Ramanujan phi function can be found here.

Source of Name

This entry was named for Srinivasa Aiyangar Ramanujan.

Sources

• 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $2$. Sums Related to the Harmonic Series or the Inverse Tangent Function
• Weisstein, Eric W. "RamanujanPhi-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujanPhi-Function.html