Sharp additive divisor sum bounds

I believe that the state-of-the-art is the result of Meurman (1999), which yields that $$S(R,h) = M(R,h) + E(R,h),$$ where $M(R,h)$ is an explicit main term, and $$E(R,h)\ll_\varepsilon (R^2+Rh)^{1/3}R^\varepsilon+ (R^2+Rh)^{1/4}R^\varepsilon\min(R^{1/4},h^{1/8+\theta/2}).$$ Here $\theta$ is an exponent admissible towards the Ramanujan-Petersson conjecture. Currently, $\theta=7/64$ is available by the work of Kim-Sarnak (2003), while the conjecture is that $\theta=0$ is also admissible. In fact the method of Motohashi (1994) can yield the same bound, as shown by Balkanova-Frolenkov (2017); see also the preprint version.

While indeed for $1\le |h|\le \sqrt{R}$ one has a uniform asymptotic formula with main term of leading order given by $$C_h R \log^2 R, \qquad C_h := \frac{1}{\zeta(2)}\sum_{d \mid h}d^{-1},$$ which is much more than the OP asked for, I'd like to complement Anurag Sahay's comment and GH from MO's answer with a discussion of lower and upper bounds (beyond $|h|\le \sqrt{R}$ and even Meurman's $|h|\le R^{2-\varepsilon}$) since the original question is about bounds.

The bottom line is that a lower bound of the expected magnitude holds for $1\le |h|\le R^{c \log \log R}$ via a quick argument while an upper bound holds for $1\le |h|\le R^A$ ($A$ arbitrary) via more intricate (yet elementary) arguments.

Lower bounds are especially easy, because $d(r)\ge \sum_{e \mid r,\, e \le \sqrt{R}} 1$, so $$\sum_{R\le r < 2R} d(r)d(r+h) \ge \sum_{e_1,\, e_2 \le \sqrt{R}} \sum_{\substack{R \le r < 2R\\e_1 \mid r,\, e_2 \mid r+h}} 1.$$ The inner sum is $R/[e_1,e_2] + O(1)$ if $(e_1,e_2)\mid h$ and $0$ otherwise. The error terms $O(1)$ sum to $O(R)$ and the main terms $R/[e_1,e_2]$ sum to $R \sum_{(e_1,e_2)\mid h,\, e_i \le \sqrt{R}}1/[e_1,e_2]$ which, by an elementary computation, equals $$R \sum_{(e_1,e_2)\mid h,\, e_i \le \sqrt{R}}\frac{1}{[e_1,e_2]}=\frac{C_h}{4} R \log^2 R + O(R E),$$ $$E := d(h) \frac{\log^2 R}{\sqrt{R}} + \log R \sum_{d \mid h} \frac{1+\log d}{d}.$$ (Some details: write $e_1=g e'_1$, $e_2=g e'_2$ for $g=(e_1,e_2)$; detect coprimality of $e'_1$ and $e'_2$ using Möbius inversion.)

As long as $d(h)=o(C_h \sqrt{R})$ and $\sum_{d \mid h} \frac{\log d}{d}=o(C_h \log R)$ this gives a lower bound of the expected size divided by $4$; this factor arises because we only considered divisors of $r$ and $r+h$ that are $\le \sqrt{R}$.

The two conditions are met in the huge range $h\le R^{c \log \log R}$: the first condition holds in this range due to Wigert's divisor bound $d(h)\le h^{(\log 2 + o(1))/\log \log h}$; for the second condition, observe that $\sum_{d \mid h} \frac{\log d}{d} \ll \log h \sum_{p\mid h} \frac{\log p}{p} \ll C_h \sum_{p \mid h} \frac{\log p}{p}$ by Mertens, so the condition is met if $\sum_{p \mid h} \frac{\log p}{p} =o( \log R)$. This bound holds in the wider range $\log \log h=o(\log R)$.

A relevant reference is

• N. Ng and M. Thom, "Bounds and conjectures for additive divisor sums", Funct. Approximatio, Comment. Math. 60, No. 1, 97-142 (2019).

See their Theorem 1.2 which establishes lower bounds for correlation of higher divisor functions $\tau_k$ when $k\ge 3$ uniformly in a similar range of $h$. Strictly speaking, $k=2$ is not covered, but the ideas are there of course.

Upper bounds are more intricate. Some context: the problem of upper bounding $\sum f(a_n)$ for general nonnegative multiplicative functions $f$ and increasing sequences $a_n$ was investigated by D. Wolke ("Multiplikative Funktionen auf schnell wachsenden Folgen", Crelle 251, 54-67 (1971)). Some special cases received special treatment. The case where $a_n$ is the indicator of an intersection of a short interval and an arithmetic progression was famously treated by Shiu. The case $f=d_k$ and $a_n = P(n)$ for $P(x) \in \mathbb{Z}[x]$ was considered by van der Corput, by Erdős and by Landreau. The choice $a_n=P(n)$ is extremely related to the OP's question because for $P(x)=x(x+h)$, $\sum d(P(n))$ almost recovers OP's sum. See T. Tao's blogpost "Erdos’ divisor bound" for a detailed exposition on the work on $\sum d(P(n))$.

The state of the art for upper bounds in your particular problem is in the paper

• K. Henriot, "Nair-Tenenbaum bounds uniform with respect to the discriminant", Math. Proc. Camb. Philos. Soc. 152, No. 3, 405-424 (2012); erratum ibid. 157, No. 2, 375-377 (2014).

A special case of his Theorem 5 (with $y=x$, $F(a,b)=d(a)d(b)$, $Q_1(x)=x$ and $Q_2(x)=x+h$) is exactly that $$(\star)\, \sum_{R\le r < 2R} d(r)d(r+h) \ll_A C_{h} R \log^2 R$$ holds for $1\le |h|\le R^A$ ($A$ arbitrary). See Theorem 6 for a matching lower bound in the same range.

His introduction mentions a few works that are superseded by his results, but the proof methods differ and may be of interest:

• R. Holowinski, "A sieve method for shifted convolution sums", Duke Math. J. 146, No. 3, 401-448 (2009). I recommend the standalone Chapter 15.6 in "Opera de Cribro" which is based on Holowinski's paper.
• Unpublished work of S. Daniel, which among other things gives $(\star)$.