Characterize the effect of absolute values on the cubic remainder

Determine whether replacing each summand of the constrained cubic remainder by its absolute value, namely studying the sum of 1/|x^3+y^3-z^3| over positive integers x,y,z with z\ne x and z\ne y, changes the logarithmic divergence of R(m,3), and characterize the resulting growth if it does.

Background

The paper expects the signed remainder R(m,3) to diverge logarithmically, with positive and negative contributions potentially canceling. Taking absolute values removes this cancellation, and the infinitely many cubic near misses with value +1 or −1 would each contribute positively. The density of these near misses could therefore alter the divergence rate, but the paper leaves this unresolved.

References

There is a separate interesting question that is worth exploring: if the absolute value of the terms in the sum R(m,3) were used instead i.e. the sum of 1/|(x+y3-z3)| with the same constraint z\ne x and z\ne y, would this modify the logarithmic divergence of R(m,3) and if so, how?

— Summing the reciprocal of the polynomial appearing in Fermat's Last Theorem  (2609.02112 - Edery, 2 Sep 2026) in Section 4, “Conclusion”