---------- Forwarded message ---------- From: Date: Sat, Dec 31, 2011 at 9:18 AM Subject: PNAS MS# 2011-19248 Decision Notification To: asills@georgiasouthern.edu Cc: asills@georgiasouthern.edu December 31, 2011 Title: "Overwhelming evidence against Rademacher's Partial Fraction Conjecture" Tracking #: 2011-19248 Authors: Sills and Zeilberger Dear Dr. Sills, I regret to inform you that the PNAS Editorial Board has rejected your manuscript [MS# 2011-19248]. The expert who served as the editor obtained 1 review, which is included below. After considering the review and re-reading the manuscript, both the editor and Board concur with the negative reviewer that the paper should be rejected. It is our policy that a single negative review should mandate rejection provided that the editor agrees with the negative review. We hope that the review will be useful to you in revising the work for submission to a more specialized journal. Once a paper has been rejected, it may not be resubmitted through an Academy member. Note that the PNAS License to Publish conveyed at initial submission is terminated. Thank you for submitting your manuscript to PNAS. I am sorry we cannot be more encouraging this time, and I hope you will consider submitting future work to PNAS. Sincerely yours, Inder M. Verma Editor-in-Chief ********************* Editor Comments: What is relevant when considering possible publication in PNAS is the quality of the results and how well the paper is written. The results strongly suggest that an old conjecture of Hans Rademacher is false, and some numerical evidence that some sort of periodicity holds for the coefficients Rademacher considered. There is a suggestion at the end of a possible way to attack the problem of showing that some of the conjectures in this paper can be proven. However, the integral proposed is a bit more complicated than the authors seem to realize. The q gamma function has one representation for |q|<1, a different one when |q|>1, and a natural boundary when |q|=1. Any circle about q=1 contains points in all three regions. My reason for not recommending publication in PNAS has to do with the results not being sufficiently developed to justify publication and I recommend the authors submit their interesting start in a different journal. Two possible journals are Experimental Mathematics or The Ramanujan Journal. Reviewer Comments: Reviewer #1: Suitable Quality?: No Sufficient General Interest?: Yes Conclusions Justified?: Yes Clearly Written?: No Procedures Described?: Yes Comments (Required): The Rademacher Conjecture is a longstanding conjecture concerning the analytic properties of the partial fraction decompositions arising from the generating function for p(n). The present article provides strong evidence that the original conjecture is false. The evidence is obtained by means of clever computation combined with computer power which is now available. The merits of the paper: 1) Although the Rademacher Conjecture is not disproved, sufficient evidence is presented to shed ample doubt. 2) Based on the numerics, the authors do more than provide evidence. They suggest a periodic phenomenon which is a major step towards figuring out the actual limiting behavior. This contribution in the paper is more substantial than 1). Indeed, the science is about figuring out the truth. 3) Whether PNAS is a proper home for this paper is the question. Here are this reviewer's thoughts. a) If FLT were false, and someone found this to be the case because they found (with the help of a computer) some numbers a,b,c and a prime p for which a^p+b^p=c^p, then would that be a PNAS paper? This referee does not think so. However, the present paper is not quite like this. b) The RH is known to be equivalent to many delicate conjectures. One could imagine a situation where a computer (with clever algorithms) tests one of these equivalent forms, and gives numerics which lead to the implausibility (but not disproof) of RH. In this case, the reviewer would strongly support a PNAS paper. Such a paper raises questions which have the potential of inspiring deep questions which lead to important discoveries. This reviewer feels that the present paper is closer in spirit to situation b). Although the Rademacher Conjecture is not as important as RH, it is central.