Summatory function ($A(x)={\sum }_{n\le x}a(n)$)

The summatory function of an arithmetic function $a$ is $A(x)={\sum }_{1\le n\le x}a(n)$. For example, the prime-counting function sums the indicator of the prime numbers, while the divisor summatory function sums the divisor function. Grouping equal values of $\lfloor x/d\rfloor$ often speeds up identities involving summatory functions.