# Backlund counting formula for Dirichlet L-functions?

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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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

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
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