Dear colleagues,
I would like to share with you my recent research, now available on Zenodo: đź”— https://zenodo.org/records/17009249
The paper investigates the normalized iterates of the sum-of-divisors function in relation to Schinzel’s conjecture, employing refined sieve methods, entropy-based analysis, and large-scale computations. The work develops an entropy deficit framework, derives polylogarithmic upper bounds on a density-one subset of integers, and explores implications for Robin’s criterion and the Riemann Hypothesis.
I am particularly seeking restrictive and technical comments from experts in analytic number theory. Critical feedback will help me refine both the heuristic and computational aspects of the approach and clarify its potential role in addressing long-standing conjectures.
Your insights would be greatly appreciated.
Best regards, Rafik Zeraoulia
Possibly — promising, but not yet a complete proof. Zeraoulia’s recent “entropy–sieve” papers (preprints from Aug 2025) present a novel route that brings new ideas to the Schinzel problem, but important, non-trivial gaps remain before one can claim a positive resolution. Bringing Shannon-entropy techniques (which have had real success in other analytic number-theory problems) into the study of multiplicative iterates is an original move and can expose structure that traditional approaches miss (Entropy methods have precedent in recent work in multiplicative number theory). The preprints show theorems that if certain vanishing-tail/shifted-prime hypotheses hold (or if one can make some analytic envelopes fully explicit), then bounded entropy would imply infinitely many small iterates — exactly the heart of Schinzel’s lim-inf statement. Zeraoulia’s entropy–sieve approach is an innovative and promising program that makes Schinzel’s conjecture look closer by reframing the problem and isolating the key analytic bottlenecks. However, as of the available preprints (Aug 2025), it stops short of an unconditional proof because it requires either (a) new effective control of tails/shifted-prime phenomena or (b) the removal/verification of certain vanishing-tail hypotheses. So it’s a plausible path, but important hard steps remain.
Nirmalya Mukhopadhyay Thank you very much for your encouraging and thoughtful comments on my work. I greatly appreciate your recognition of the potential of the entropy–sieve approach and your careful assessment of the remaining challenges.
I would like to inform you that I have already uploaded a new and significantly improved version of the paper, in which I have completed several of the previously missing proofs and clarified the analytic bottlenecks you mentioned. This updated version provides a more rigorous foundation for the results and addresses some of the gaps highlighted in the earlier preprint.
I truly value your feedback, and I hope the new version better reflects the scope and strength of the method. I would be delighted to hear any further thoughts you might have.
Thanks for reading my comments carefully. I must appreciate your research efforts. Many congratulations on your new version of the paper. Best wishes.
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