Average orders of multiplicative functions

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

A simpler construction is possible: you can just use the greedy algorithm to get close to the desired density. For example, selecting primes as needed to keep the density of squarefree numbers composed of just those primes yields

2, 3, 11, 17, 29, 31, 47, 61, 79, 83, 113, 127, 131, 137, 149, 157, 173, 211, 223, 227, 233, 269, 281, 293, 307, 331, 337, 373, 401, 409, 421, 439, 443, 509, 547, 557, 577, 593, 599, 607, 613, 631, 653, 659, 733, 739, 757, 761, 787, 797, 839, 863, 937, 953, 967, 977, 983, 1009, 1087, 1091, 1093, 1109, 1151, 1153, 1163, 1171, 1181, 1213, 1259, 1289, 1297, 1303, 1409, 1423, 1429, 1439, 1453, 1499, 1543, 1549, 1567, 1597, 1627, 1637, 1657, 1667, 1709, 1721, 1723, 1787, 1823, 1861, 1867, 1873, 1951, ...

and through 10^7 the sum of the indicator function is within 14 of the desired density x/log x.