Transcendental equations satisfied by the individual zeros of Riemann $\zeta$, Dirichlet and modular $L$-functions

Abstract

We consider the non-trivial zeros of the Riemann ζ-function and two classes of L-functions;Dirichlet L-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer n. From this it follows that the ordinate of the n-th zero satisfies a transcendental equation that depends only on n. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert W -function. Our approach is a novel and simple method, that takes into account arg L, to numerically compute non-trivial zeros of L-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the ζ-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet L-functions and the L-function for the modular form based on the Ramanujan τ -function, which is closely related to the bosonic string partition function. *

Some Riemann Hypotheses from random walks over primes

“…The above equation is actually also valid for ζ(s) and other principal L-functions. We argued in [12] that if the above equation has a solution for every n, then it saturates the known counting formula on the entire strip, implying that all zeros must be on the critical line. There is a unique solution for every n if one ignores the arg L term, which can be expressed in terms of the Lambert-W function to a very good approximation.…”

“…Consider zeros ρ n = 1/2 + it n of non-principal and primitive Dirichlet L-functions. In [12] we derived the following transcendental equation on the critical line without assuming the Riemann Hypothesis: where n = 1, 2, . .…”

“…(57)]. Although not discussed in [12], it is now clear that this only applies to cusp forms. For non-cusp forms the transcendental equation will not have a solution.…”

On the explicit upper and lower bounds for the number of zeros of the Selberg class

“…There are also explicit upper bounds for the number of the zeros for the Riemann zeta-function assuming Riemann hypothesis [2] and along the critical line [20] and L-functions [3]. G. França and A. LeClair [9] proved an exact equation for the nth zero of the L-functions on the critical line. There are also explicit results for Dirichlet L-functions and Dedekind zeta-functions, see for example the papers from K. S. McCurley [12] and T. S. Trudgian [21].…”

Generalized Riemann hypothesis, time series and normal distributions

“….. 2 . An explicit formula for the n − th zero as the solution of a transcendental equation was proposed in [57].…”