## THE ZETA FUNCTION ON THE CRITICAL LINE: NUMERICAL EVIDENCE FOR MOMENTS AND RANDOM MATRIX THEORY MODELS

Citations: | 2 - 1 self |

### BibTeX

@MISC{Hiary_thezeta,

author = {Ghaith A. Hiary and Andrew and M. Odlyzko},

title = {THE ZETA FUNCTION ON THE CRITICAL LINE: NUMERICAL EVIDENCE FOR MOMENTS AND RANDOM MATRIX THEORY MODELS},

year = {}

}

### OpenURL

### Abstract

Abstract. Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing 109 zeros near zero number 1023. More than anything else, the variability illustrates the limits of what one can learn about the zeta function from numerical evidence. It is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws. Also, some long range correlations in the values of the second moment, as well as asymptotic oscillations in the values of the shifted fourth moment, are found. The computations described here relied on several representations of the zeta function. The numerical comparison of their effectiveness that is presented is of independent interest, for future large scale computations. 1.

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Citation Context ...1.3) encounter a problem, which we illustrate in the case of the second moment, known to satisfy (1.7) ∫ T 0 |ζ(1/2 + it)| 2 dt = T log T 2π + T (2γ − 1) + E1(T ) , where E1(T ) = O(T 35/108+ɛ ); see =-=[I2]-=-. (This result does not depend on the assumption of the Riemann hypothesis.) Based on (1.7), one might suspect (1.8) 1 H ∫ T +H T |ζ(1/2 + it)| 2 dt = T + H H T + H log 2π However, in order for (1.8) ... |

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Citation Context ...ζ(1/2+it)| T 2 |ζ(1/2 + it + iα)| 2 dt ≈ ( ∫ T +H 1 H T |ζ(1/2 + it)| 2 dt ) ( 1 H ∫ T +H T |ζ(1/2 + it + iα)| 2 dt and the latter is directly related to the autocovariances. Kösters [Ko] and Chandee =-=[Ch]-=- investigated the function M(T, T ; α) as well as other more general shifted moments. Kösters’ work immediately implies if α log T = O(1) (as T → ∞), and if (4.15) K(T ; α) := then 12 (α log T ) 2 ( 1... |

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Citation Context ...ind similar evidence for such correlations in the case of higher moments. To further examine the correlations in the second moment, we numerically investigate the shifted fourth moment, where Kösters =-=[Ko]-=- proved a kernel law for shifts on the scale of mean spacing of zeros. For larger shifts, we observe a departure from this law, and the onset of asymptotic oscillations; see Figures 3 and 4. In the co... |

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Citation Context ...by Hardy, Littlewood, Selberg, Titchmarsh, and many others. It has long been conjectured that the 2kth moment of |ζ(1/2 + it)| should grow like ck(log t) k2 for some constant ck > 0. Conrey and Ghosh =-=[CG1]-=- reformulated this long-standing conjecture in a precise form: for a fixed integer k > 0, and as T → ∞, (1.1) ∫ T 1 |ζ(1/2 + it)| T 0 2k dt ∼ a(k)g(k) k2 (log T ) ! k2 , where a(k) is a certain, gener... |

1 |
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Citation Context ...nd similar evidence for such correlations in the case of higher moments. To further examine the correlations in the second moment, we numerically investigate the shifted fourth moment. There, Kösters =-=[Ko]-=- proved a kernel law for shifts on the scale of mean spacing of zeros. For larger shifts, we observe a departure from this law, and the onset of asymptotic oscillations; see Figures 3 and 4. In the co... |