Maximum-so-far

Definition

Suppose $f$ is an arithmetic function from the natural numbers to a subring of the ring of real numbers. Then, a natural number $n$ is termed a maximum-so-far for $f$ if $f(n)\ge f(m)$ for all natural numbers $m\le n$.

Related notions

• Strict minimum-so-far is a natural number $n$ such that $f(n)<f(m)$ for $m<n$.
• Minimum-so-far is a natural number $n$ such that $f(m)\ge f(n)$ for all $m\le n$.
• Strict maximum-so-far is natural number $n$ such that $f(m)<f(n)$ for all $m<n$.