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