Characterize the profiles that are functions of good variation

Characterize the recurrence-admissible positive profiles that are functions of good variation, particularly those that are bounded and Riemann integrable or slowly varying at the origin, by proving or refuting the stated membership conjecture.

Background

A function of good variation is a profile whose discrete equation has a finite transparency threshold and whose partial sums obey a common absorption estimate above that threshold. The volume proves membership for several explicit families, including affine and selected logarithmic profiles, but not in the general positive bounded or slowly varying classes.

The conjecture excludes degenerate profiles such as the diagonal profile, for which no exponent is transparent. Its proof would provide a general existence theory for the regularity index within the stated profile classes.

References

The conjecture is not proved in the present volume, and the general membership question is open.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Conjecture 2.8 and its proof-status note, Section 2.5

The weaker hypothesis \int_01g(t)\,dt\neq0 would cover profiles that change sign, and the question of whether it suffices is open.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Remark ‘A stronger form, and what it would need’, Section 2.5

Is \gamma_{1/2,0} the leftmost zero of *{1/2,0}, so that \eta(g{1/2,0})=\gamma_{1/2,0}? A numerical computation finds a pair of zeros near 0.3842\pm1.1016i, and a numerical search finds none with Re z\le\gamma_{1/2,0}, which is not a proof that none exists.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Open Problem ‘The original lacunary profile’, Section 5.6

Which arithmetic functions admit a counting formula whose count is slowly varying with bounded increments is not answered here, and the two cases above are the only ones this volume records.

— Regular Arithmetic Functions, Volume I. Theory, Applications, Examples  (2609.09366 - Cloitre, 8 Sep 2026) in Section 3.4, after the two counting examples

For distinct interior Green poles and for model branches one has $b_0=0$, so the two inward half-line derivatives sum to zero. At a coalescing-pole parameter, an atom at $0$ may occur and the derivative sum need not vanish. Atomlessness alone leaves open the possibility of a singular continuous part in $u_h''$.

— A sharp two-disk bound for the second positive Neumann eigenvalue under a curvature upper bound  (2609.26609 - Liu et al., 22 Sep 2026) in Section 3, subsection “The abstract kernel and its minimizer,” paragraph following Proposition [Positive kernel and normalized minimizing profile]