Over 40 years ago I was fascinated by analytic number theory. I think it sprung from being gifted as a teen a copy of Ornstein Ore’s book, Number theory and it’s history. In any case, I was focused on learning as much about the Riemann zeta function as I could. It was amazing to me then and now how much number-theoretic information a single function like that could encode. The non-trivial zeros of are supposedly all on the line . Using arguments of A. Fujii (On the difference between r consecutive ordinates of the zeros of the Riemann zeta function, Proc. Japan Acad., 1975) and A. Ghosh (On the Riemann zeta function—mean value theorems and the distribution of |S(t)|, J. Number Theory, 1983), I wrote a preprint A note on gaps between zeros of L-functions (pdf here) to understand a conjecture of Hugh Montgomery. Years later I generalized the result from to any Dirichlet series with several properties analogous those of (such as an Euler product and so on), using the setup I had in the first book I wrote, Distribution theorems of L-functions (Pitman Research Notes,1986).

I don’t remember if I submitted it and it got rejected, or if I just posted it on my faculty webpage and forgot about it. In any case, it was never published. But, with the presence of AI as a math “paper referee”, I asked chatgpt and claude to be referees. They were asked to read it and suggest corrections, if any, to imprecise mathematical arguments, or revisions for readability. This process when through several revisions, with one or the other acting as the referee for various revisions. This is the AI-aided corrected version (pdf here).

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