# Average orders of multiplicative functions

For a multiplicative function $f$ and $x>0$ let $$S_f(x)= \sum_{n \leq x} f(n).$$ Studying sums of this type is a favourite pastime of analytic number theorists. I'm trying to understand what kind of behaviour can occur for such sums. In particular, my question is the following.

Does there exist a multiplicative function $f$ and a constant $c_f>0$ such that $$S_f(x) \sim c_f\frac{x}{\log x},$$ as $x \to \infty$?

Here is some motivation for how I came across this problem. Analytic number theorists often study sums of the above type by studying the analytic properties of associated Diriclet series $$L(f,s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}.$$

Here, if $L(f,s)$ has a pole of order $r>0$ at $s=1$ and is well-behaved for $\text{Re}( s) >1$, then one can often show (using e.g. a Tauberian theorem such as Perron's formula) that we have an asymptotic formula $$S_f(x) \sim c_f x (\log x)^{r-1}.$$

More generally there is the Selberg-Delange method, here one works with complex powers $\zeta^z(s)$ of the Riemann zeta function. This method, when it works, will give an asymptotic formula of the shape $$S_f(x) \sim c_f x (\log x)^{z-1}.$$ In particular, it does not seem that one can obtain an asymptotic formula like the one I am seeking using this approach.

Note that one cannot use something like the prime number theorem to construct an example of the shape I am looking for, since $\pi(n)$ is not a multiplicative function!

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One possibility for constructing such a function is to find a multiplicatively closed set $A$ (meaning that $m,n\in A$ if and only if $mn\in A$) whose counting function satisfied $\#\{n\le x\colon n\in A\} \sim cx/\log x$. Then you could simply take $f$ to be the indicator function of $A$. –  Greg Martin Dec 5 '12 at 4:45
@Greg: Do you know an example of such a set? –  Daniel Loughran Dec 5 '12 at 9:47
Not offhand, no.... –  Greg Martin Dec 6 '12 at 10:20
Hi Daniel, interesting question. It seems that at least one nice thing that can be said is that the primes in such a multiplicatively closed set must have zero density in the primes. This would be due to Theorem 1 of this paper (arxiv.org/abs/1110.0708) of Moree, and also from this MO question (mathoverflow.net/questions/94543/density-of-a-set-of-integers), Density of a set of integers. Wonder exactly how sparse they have to be, though. –  Timothy Foo Dec 6 '12 at 22:33

I think an explicit multiplicative function that should do the job for you is this one: $f(2^n)=2^n/\big((n+1)\sqrt{\log(n+e)}\big)$, $f(3^n)=3^n/\big((n+1)\sqrt{\log(n+e)}\big)$, $f(p^n)=0$ for $n\ge 1$ and primes $p\ge 5$. The $+1$'s and $+e$'s are just there to make the formula make sense for $n=0$ also.

Here's an outline of why: It suffices to show that for all $\epsilon>0$, for all sufficiently large $N$ one has $$\sum_{N < 2^k3^l\le (1+\epsilon) N} \frac{1}{(k+1)\sqrt{\log(k+e)}}\cdot \frac{1}{(l+1)\sqrt{\log(l+e)}}\sim c\epsilon /\log N$$ Taking logs, the summation range is essentially $\log N\le k\log 2+l\log 3\le \log N+\epsilon$. For large $N$, there are approximately $\epsilon\log N/(\log 2\log 3)$ pairs $(k,l)$ in the range. These are reasonably uniformly distributed in the band of $\mathbb R^2$, $x\log 2+y\log 3\in [\log N,\log N+\epsilon]$. Hence (with a bit of work making sure the ends of the integral don't dominate), the sum is close to $$\frac{\epsilon}{\log 2}\int_{0}^{\log N/\log 3} \frac{1}{(y+1)\sqrt{\log (y+e)}}\frac{1}{(x+1)\sqrt{\log(x+e)}}\ dx,$$ where $y=(\log N-x\log 3)/\log 2$.

In the first half of the range, the integral is something like $$\frac{\epsilon}{\log N\sqrt{\log\log N}} \int_0^{\log N/(2\log 3)} \frac{1}{(x+1)\sqrt{\log(x+e)}}\ dx$$ which is $\sim c\epsilon/\log N$. Similarly for the second half of the range.

If you prefer your multiplicative functions to be integer-valued, of course, you can just take the floor of everything.

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I think this works. Very nicely conceived. –  Greg Martin Dec 7 '12 at 3:27

You can find a many such functions in the book Sándor, J.; Mitrinović, D. S. & Crstici, B. Handbook of number theory. I Springer, 2006 (see for example ch. IV). Some asymptotic formulae have the form $\frac{x^2}{\log x}$ but you can easely transform them in the form $\frac{x}{\log x}$ using summation by parts.

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Is it possible to construct something from the ground up? Just say, I prescribe $f(p^k) = A_{pk}$ and have the average order come out right? –  Will Jagy Dec 5 '12 at 1:55
Alexey: I don't see how you can produce $\frac{x}{\log x}$ from $\frac{x^2}{\log x}$. Summation by parts destroys multiplicativity, I think. –  GH from MO Dec 5 '12 at 4:46
If the sum of $a(n)$ grows like $\frac{x^2}{\log x}$ then the sum of $a(n)/n$ grows like $\frac{x}{\log x}$. – Alexey Ustinov 25 mins ago –  Alexey Ustinov Dec 5 '12 at 9:02
@Alexey: Thanks for the answer. I will try to find this book in our library and get back to you. Is it possible for you to give a simple example in the mean time? –  Daniel Loughran Dec 5 '12 at 21:02
Chapter IV of this book is a compendium of results related to four non-multiplicative functions (the largest prime divisor, the smallest prime divisor, and the sum of the prime divisors, counted with or without multiplicity). Now there may still be a result in this chapter that fits the bill, but maybe a more precise reference could be given? –  Anonymous Dec 5 '12 at 21:19