-1

On a recent answer about arithmetic functions, I often use quantities like $F(\lfloor n/i \rfloor)$, where $F$ is the summatory function of $f$, i.e. $F(n) = \sum_{i \le n} f(i)$.

Is it cleaner to extend $F$ to the real domain by defining $F(x) = \sum_{i \le x} f(i)$, equivalently $F(x) := F(\lfloor x \rfloor)$? This lets me leave off lots of floor symbols that may be distracting, but when I want to do counting for how many $i$ have equal $F(n/i)$ because the input is actually floored, I have to introduce them again, so it could be more confusing going back and forth. What I can also do is specify all divisions as floor divisions, but this likely serves to make things even more confusing.

qwr
  • 11,362
  • If you're asking if $$\sum\limits_{i=1}^{\lfloor x\rfloor} f(i)=\sum\limits_{i=1}^x f(i)$$ then yes this is true. Many functions such as Mertens function $$M(x)=\sum\limits_{n=1}^x \mu(n)$$ are typically defined without use of the floor function. I don't understand your comment about having to reintroduce the floor function. – Steven Clark Aug 01 '24 at 16:36
  • @StevenClark it is shown in my linked answer. Basically for my simplified counting of the Dirichlet hyperbola method, there is a term like F(floor(n/i)), and for many large values of i this argument floor(n/i) turns out to be the same. So I can write F(n/i) but for precise counting I need to reintroduce the floor. – qwr Aug 01 '24 at 16:44

1 Answers1

0

Yes, it is very common to use functions like $$ F(x) = \sum_{n\le x} f(n) $$ when summing arithmetic functions. Such summatory functions take a real number as input, and they are step functions—constant on every interval $[n,n+1)$ with jump discontinuities of size $f(n)$ at $x=n$ (unless $f(n)=0$). (It's also usually implicit that the sum is over only positive integers $n$.)

The notation $\sum_{n\le x}$ is preferred to $\sum_{n=1}^x$, since the latter notation has connotations that the upper bound should be an integer.

Greg Martin
  • 92,241
  • Do you think in the context of my linked answer if would reduce confusion? Or is it clearer to be explicit with all the floors? Or does it not matter. – qwr Aug 01 '24 at 17:22