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Let $f(n)$ be a nonnegative arithmetic function satisfying

  1. $f(p^l) \leq A_1^l$ for all primes $p$, integers $l\geq 1$, and some constant $A_1$;
  2. $f(n) \leq A_2 n^\varepsilon$ for all $\varepsilon > 0$ and some constant $A_2 = A_2(\varepsilon)$.

From the work of Shiu, we know that for functions $f$ satisfying 1. and 2., we have the estimate $$ \tag{1} \sum_{x-y<n\leq x} f(n) \ll \frac{y}{\log x} \exp\left(\sum_{p\leq x} \frac{f(p)}{p}\right) $$ uniformly for $x^\alpha < y < x$ and any fixed $\alpha \in (0,1)$.

I'd like to know if such a bound has been proved for functions like $$ f(n) = \sum_{d^2\mid n} d^\theta, $$ where $0 <\theta < 1$. This function fails both conditions 1. and 2. Nevertheless, it is bounded on average. One can use Dirichlet's hyperbola method to establish (for some constant $C = C(\theta)$) $$ \sum_{n\leq x} \sum_{d^2\mid n} d^\theta = \zeta(2-\theta)x + Cx^{\frac{1+\theta}{2}} + O\left(x^{\frac{1+\theta}{3}}\right), $$ which immediately gives a result like (1) in the range $y \geq x^{\frac{1+\theta}{3}}$. Certainly a result like (1) cannot hold in a range like Shiu's work; indeed, for any positive integer $m$, take $x = m^2$ to see that $$ \sum_{x - y < n\leq x} \sum_{d^2\mid n} d^\theta \geq m^\theta = x^{\frac{\theta}{2}}, $$ so (1) fails for $y \ll x^{\frac{\theta}{2}}$. I conjecture that this is the only obstruction, so a result like (1) should hold for $y > x^{\frac{\theta}{2}+\alpha}$ and any $\alpha \in (0,1)$.

This certainly seems like the thing someone might have considered before, but I'm not familiar enough with the literature to know. Any references, ideas, or feedback would be most appreciated.

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  • $\begingroup$ I believe one can improve the upper bound, though not all the way to the conjecture, using the determinant method. View the problem as counting pairs $c,d$ with $x-y<cd^2 \leq x$. One can check by computing a determinant that if three pairs $(c,d)$ are close enough together (say, in the $d$ coordinate), then they are colinear. So we can divide the range in $d$ into intervals where in each interval all solutions lie on a line or there's not many solutions. Then bounding the number of solutions on a line is bounding the number of times a one-variable cubic polynomial takes values in a range. $\endgroup$
    – Will Sawin
    Commented Oct 13 at 18:52
  • $\begingroup$ @WillSawin Oh interesting. Hmm, do you have a good reference/recommended reading for the determinant method? I've heard it mentioned, but I'm not familiar with it myself. $\endgroup$
    – Joshua Stucky
    Commented Oct 13 at 19:35

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I realized this is actually pretty much the same problem as one arising when studying squarefree numbers in short intervals. Using some results from Filaseta and Trifonov [1] (specifically the estimate (4) in that paper), it's not too hard to show that $$ \sum_{x - y < n \leq x} \sum_{d^2\mid n} d^\theta \ll_\theta y $$ for $\theta < \frac{1}{3}$ and $y > x^{\frac{1+\theta}{4}}\sqrt{\log x}$. One can probably improve the range of $\theta$ and $y$ by using all of the ideas in that paper, perhaps to something like $\theta < 1$ and $y > x^{\frac{1+\theta}{5}+\varepsilon}$.

Their method uses first and second differences, essentially coming from Pade approximations. This leads to calculating the determinant of a matrix of polynomials. If I'm not mistaken then, it seems that their method is really an application of the determinant method, but where their auxiliary polynomials arise from Pade approximations.

[1] Filaseta, Michael; Trifonov, Ognian, On gaps between squarefree numbers. II, J. Lond. Math. Soc., II. Ser. 45, No. 2, 215-221 (1992). ZBL0799.11032.

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