On the Summatory Function of $d_3(n)$

Abstract: In this article, we refine the method of our earlier work with N. Paloj{ä}rvi to obtain a sharper explicit bound for the error term $Δ<em>{3}(x)$ associated with the summatory function of $d</em>{3}(n)$. We prove that \begin{equation*} |Δ_3(x)| < \begin{cases} 0.6901\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 3.682\cdot 10^{31}\le x < 4.133\cdot 10^{87},\[4pt] 0.2067\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 4.133\cdot 10^{87} \le x < 1.597\cdot 10^{98},\[4pt] 0.1947\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & x \ge 1.597\cdot 10^{98}. \end{cases} \end{equation*} These explicit results improve the exponent of $x$ from $2/3$, due to Tudzi, and $859/1400$, due to Paloj{ä}rvi and Tudzi, to $1/2$, giving the best known bound for all $x\ge 3.682\cdot 10<sup>{31}$.

• The paper improves the error term bound in the summatory function by reducing the exponent from 2/3 to 1/2 through explicit analytic techniques.
• It refines Perron’s formula and leverages explicit third moment estimates of ζ(s) to optimize the bound with fully specified logarithmic terms.
• The paper establishes effective constants across vast ranges of x, setting a new benchmark for explicit results in additive divisor problems.

Explicit Bounds for the Error Term in the Ternary Divisor Problem

Introduction and Context

The paper addresses a central problem in analytic number theory: obtaining explicit estimates for the error term in the asymptotic formula for the summatory function of the ternary divisor function $d_3(n)$. This function, $d_3(n)$, counts the number of ordered triples of positive integers whose product is $n$. The summatory function, $T_3(x) = \sum_{n \leq x} d_3(n)$, admits the classical decomposition:

$T_3(x) = xP_3(\log x) + \Delta_3(x)$

where $P_3(\log x)$ is a degree-2 polynomial with explicit coefficients, and $\Delta_3(x)$ is the error term. Improving the quantitative understanding of $\Delta_3(x)$ is linked to moments of the Riemann zeta function, and, in the general case, to the broad "Dirichlet divisor problem."

Traditionally, explicit bounds on $\Delta_3(x)$ were derived from Dirichlet’s hyperbola method or complex-analytic techniques, with the sharpest known explicit exponents being $2/3$ (Tudzi) and $d_3(n)$0 (Palojärvi & Tudzi), before the improvements made here.

Main Results

The principal achievement of the paper is the derivation of improved, fully explicit upper bounds for the error term $d_3(n)$1, valid for all $d_3(n)$2. Specifically, for different intervals of $d_3(n)$3, the following bounds are obtained:

$d_3(n)$4

This represents the explicit realization of the conjectural exponent $d_3(n)$5, at the cost of a logarithmic loss, across all $d_3(n)$6 in these large ranges. The prior best explicit bounds, especially over large $d_3(n)$7, were strictly weaker in the exponent.

Key claims:

• The exponent of $d_3(n)$8 in the explicit bound for $d_3(n)$9 is lowered from $n$0 (and later $n$1) to $n$2, which is the best-known unconditional result for all sufficiently large $n$3.
• The explicit constants and logarithmic powers are made fully effective via a careful mixture of analytic techniques.

Methodological Advances

The improvements are obtained via a refinement of the analytic approach initiated in earlier works. The main technical ideas involve:

• Refinement of Perron's Formula: Reduction of the error analysis to estimating truncated vertical integrals involving $n$4, for $n$5 in the critical strip.
• Optimal Use of Zeta Moments: Unlike earlier works relying mainly on the second moment of $n$6, this paper systematically exploits explicit third moment estimates for $n$7. The third moment allows one to obtain bounds corresponding to the exponent $n$8, aligning with the Lindelöf hypothesis in the convexity regime.
• Enhanced Convexity Bounds: Effective pointwise and convexity-type bounds for $n$9 are deployed for $T_3(x) = \sum_{n \leq x} d_3(n)$0 ranging from $T_3(x) = \sum_{n \leq x} d_3(n)$1 up to just above $T_3(x) = \sum_{n \leq x} d_3(n)$2.
• Explicit Computation and Optimization: All constants are specified, logarithmic compensation for the exponent gain is tightly quantified, and arithmetic complications related to integer and half-integer $T_3(x) = \sum_{n \leq x} d_3(n)$3 are separately optimized.

The proof strategy consists of first expressing the error term via shifted complex integrals; then, by leveraging explicit analytic continuation, convexity, and interpolation bounds on the zeta function, together with precise estimates of divisor sums, all contributing error terms are quantitatively analyzed.

Numerical and Technical Implications

The achievement of bounding $T_3(x) = \sum_{n \leq x} d_3(n)$4 by $T_3(x) = \sum_{n \leq x} d_3(n)$5 for enormous ranges of $T_3(x) = \sum_{n \leq x} d_3(n)$6—with a continuous improvement in the explicit constant for larger $T_3(x) = \sum_{n \leq x} d_3(n)$7—provides the definitive explicit bounds known, strictly improving the earlier $T_3(x) = \sum_{n \leq x} d_3(n)$8 and $T_3(x) = \sum_{n \leq x} d_3(n)$9 results. The logarithmic factor remains a bottleneck, intrinsic to current knowledge of zeta moments.

The technical elaboration of bounding integrals on lines $T_3(x) = xP_3(\log x) + \Delta_3(x)$0 with $T_3(x) = xP_3(\log x) + \Delta_3(x)$1, and careful management of the interplay between pointwise and mean-value theorems for $T_3(x) = xP_3(\log x) + \Delta_3(x)$2, sets a template for further work on higher $T_3(x) = xP_3(\log x) + \Delta_3(x)$3 and possibly other additive divisor problems.

The explicit nature of the results is of practical importance for computational applications, e.g., in analytic estimation for number-theoretic algorithms or the study of the Riemann zeta function’s value distribution.

Theoretical Implications and Future Directions

While the exponent $T_3(x) = xP_3(\log x) + \Delta_3(x)$4 matches the heuristic and conjectural prediction from the Lindelöf Hypothesis (the ultimate conjecture being that $T_3(x) = xP_3(\log x) + \Delta_3(x)$5), removing the superfluous logarithmic factor would unconditionally push the bound towards conjectural optimality. However, this would require progress on mean values of zeta, notably the third moment.

These techniques are readily generalizable to higher $T_3(x) = xP_3(\log x) + \Delta_3(x)$6, where the relevant moment of $T_3(x) = xP_3(\log x) + \Delta_3(x)$7 plays a corresponding role. Thus, ongoing improvements on moment bounds or the development of mean-value theorems, as well as better convexity/power-saving estimates for $T_3(x) = xP_3(\log x) + \Delta_3(x)$8, would immediately yield sharper explicit divisor error bounds.

The results also reinforce the central relationship between additive divisor problems and the complex moments of $T_3(x) = xP_3(\log x) + \Delta_3(x)$9, providing a valuable testbed for the verification and optimization of explicit complex-analytic estimates.

Conclusion

The paper establishes the sharpest available explicit bounds for the error term in the summatory function of the ternary divisor function, lowering the exponent in the main term of the bound to $P_3(\log x)$0 while maintaining effective control over logarithmic terms and explicit constants across wide ranges of $P_3(\log x)$1. The principal technical innovation is the synthesis of explicit third-moment and convexity bounds for $P_3(\log x)$2. The methodology sets a new benchmark for explicit results in higher-divisor problems and effectively links progress in the explicit analytic theory of the Riemann zeta function directly with classical problems in additive number theory ["On the Summatory Function of $P_3(\log x)$3" (2607.10053)].