Random matrices, generalized zeta functions and self-similarity of zero distributions

There is growing evidence for a connection between random matrix theories used in physics and the theory of the Riemann zeta function and L-functions. The theory underlying the location of the zeros of these generalized zeta functions is one of the key unsolved problems. Physicists are interested because of the Hilbert–Polya conjecture, that the non-trivial zeros of the zeta function correspond to the eigenvalues of some positive operator. To complement the continuing theoretical work, it would be useful to study empirically the locations of the zeros by different methods. In this paper we use the rescaled range analysis to study the spacings between successive zeros of these functions. Over large ranges of the zeros the spacings have a Hurst exponent of about 0.095, using sample sizes of 10 000 zeros. This implies that the distribution has a high fractal dimension (1.9), and shows a lot of detailed structure. The distribution is of the anti-persistent fractional Brownian motion type, with a significant degree of anti-persistence. Thus, the high-order zeros of these functions show a remarkable self-similarity in their distribution, over fifteen orders of magnitude for the Riemann zeta function! We find that the Hurst exponents for the random matrix theories show a different behaviour. A heuristic study of the effect of low-order primes seems to show that this effect is a promising candidate to explain the results that we observe in this study. We study the distribution of zeros for L-functions of conductors 3 and 4, and find that the distribution is similar to that of the Riemann zeta functions.

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