# Backlund counting formula for Dirichlet L-functions? [closed]

Are there published works on the analog of Backlund's counting formula for Riemann zeros on the strip involving Riemann-Siegel theta, but for Dirichlet L-functions? We found papers with the analog of the Riemann-van Mangoldt asymptotic version due to Selberg, but not the more exact Backlund form involving \arg \zeta.

The reason I am asking is that we have a novel way to derive such formulas, and actually have much more. We showed that zeros on the critical line are in one-to-one correspondence with zeros of the cosine function, and are thus enumerated by an integer n. They can thus be easily counted. We derived an exact equation for the n-th zero that depends on n. There is a small correction in comparison with Selberg's result, involving the arg of the Gauss sum G for the Dirichlet character, which is actually an improvement, as can be verified numerically. It is available on the arXiv, math.NT

Any useful comments would be appreciated.

Sincerely, André LeClair

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## closed as off-topic by Daniel Loughran, Lucia, Alex Degtyarev, Joonas Ilmavirta, Dima PasechnikMay 7 at 7:28

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Did you look at Chapter 16 of Davenport's book on multiplicative number theory? The details are not given but the argument is standard. –  Lucia Sep 29 '13 at 4:02
Thank you Lucia, I will have a look. But I suspect our derivation is different, since we are not deriving counting formulas on the strip, which I imagine involves the standard contour integral, but on the critical line. But I will definitely have a look, thanks. –  André LeClair Sep 29 '13 at 4:44
We've recently had some discussion of questions about preprints and their suitability. See meta.mathoverflow.net/questions/927/…, and the older discussions on tea linked from there. –  Scott Morrison Oct 2 '13 at 3:37
Scott, I am new to this site, but I see the point, and removed the link to the math.NT submission. –  André LeClair Oct 2 '13 at 4:07
I'm voting to close this question as off-topic because it's no longer relevant. –  Daniel Loughran May 6 at 22:44

We found the answer to the question in Montgomery's book, Multiplicative Number Theory. It was very useful, since our own formula, which was derived very differently, agrees exactly with Montgomery's, whereas there was a small discrepancy with Selberg's version.

This probably warrants a new question, but we are left with this one: We derived our formula by first deriving an exact formula the n-th enumerated zero on the critical line, which as far as we know is new. With such a formula, one can clearly compute a counting formula for the zeros on the critical line,
N_0(T). We discovered that it exactly agrees with Montgomery's formula N(T) for the zeros on the entire strip, N_0(T) = N(T). Can this be the basis for a proposal to demonstrate the validity of the (generalized) Riemann hypothesis, assuming the derivation of our formula for the n-th zero is correct?

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I apologize if perhaps the answer is too obvious, but it is just a simple follow-up question, which I do not know the answer to. –  André LeClair Oct 2 '13 at 3:03
Your alleged proof does not include GRH as a corollary; rather, your claim is that the number of zeroes off the line is $o(T\log T)$. This, however, would nevertheless be a huge breakthrough result, as for example for $\zeta(s)$ it is only currently known that $\limsup_{T\to\infty}N_0(T)/N(T)\geq 0.42$, or heuristically that "at least 42% of the zeroes of $\zeta(s)$ lie on the critical line", whereas you claim 100%. From a cursory glance of your paper, however, I have my doubts as there seems to be no "heavy lifting" (though it is hard to read as the notation is often nonstandard). –  Peter Humphries Oct 2 '13 at 3:13
Thank you Peter. We are not trying to make huge claims, nor alleged proofs. We are just looking for feedback. I think we have a novel way of computing N_0(T), which does not involve the usual contour integrals for N(T); how could it? Perhaps our derivation of N_0(T) is faulty, but after communicating with a number of mathematicians, we cannot pinpoint an error. Of course, I would be happy if someone indeed pointed out the error. Everyone is skeptical of course, but I do hope someone will take the time to find the error, or not. –  André LeClair Oct 2 '13 at 3:24
@PeterHumphries: Forsooth. Regrettably, the whole context is too denatured ... I've looked at earlier and current versions... –  paul garrett Oct 2 '13 at 3:24
To Andre LeClair: I'm not sure, but one issue could be the definition of the argument of the L-function in your paper (equation 8) and the usual definition. In your paper I don't see a definition, but my guess is that the argument is just varying continuously as you move on the critical line (except at zeros where it has a bump). The usual definition is to vary continuously from $2$ to $2+iT$ and then to $1/2+iT$. The definitions will match if GRH is true. Since GRH is true up to large height, I think you are just noting that equality. Just a guess -- I haven't read your paper carefully. –  Lucia Oct 2 '13 at 4:19