Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

We shortly review recent work interpreting the quotient ζ(s − 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group we set inductively Gk: = (Z/2Z) k, with Z/2Z = ({0, 1}, +). h0: = 1, hk+1(σ, 0): = hk(σ) and hk+1(σ, 1): = hk(σ) + hk(1 − σ), (1) where σ = (σ1,..., σk) ∈ Gk and 1 − σ: = (1 − σ1,..., 1 − σk) is the inverted configuration. The sequences hk(σ) of integers, written in lexicographic order, coincide with the denominators of the modified Farey sequence. We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk: = ln(hk). Thus we may interpret

Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column for July 1974 that “every recreational mathematician should buy a copy forthwith. ” That book contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (or OEIS) [24] contains 117000 sequences.

The EKG or electrocardiogram sequence is defined by a(1) = 1, a(2) = 2 and, for n # 3, a(n) is the smallest natural number not already in the sequence with the property that gcd{a(n - 1), a(n)} > 1. In spite of its erratic local behavior, which when plotted resembles an electrocardiogram, its global behavior appears quite regular. We conjecture that almost all a(n) satisfy the asymptotic formula a(n) = n(1 + 1/(3 log n)) + o(n/ log n) as n # #; and that the exceptional values a(n) = p and a(n) = 3p, for p a prime, produce the spikes in the EKG sequence. We prove that {a(n) : n # 1} is a permutation of the natural numbers and that c 1 n # a(n) # c 2 n for constants c 1 , c 2 . There remains a large gap between what is conjectured and what is proved. 1.

The computer results of the investigation of the number h_N(d) of gaps of the length d between two consecutive primes smaller than N are presented. The computer search was done up to N = 2^44 ≈ 1.76 × 10^13. The oscillations of h_N(d) are found; the most eminent have the period 6. A few attempts to fit the obtained data by an analytical formula are given. As the applications two formulae are obtained: for the largest gap between consecutive primes below a given bound and the formula for "champions" -- the most often occurring pairs of primes. Obtained data supports the conjecture that the number of Twins and primes separated by a gap of the length 4 ("Cousins") is almost the same and it determines a fractal structure on the set of primes. The possible relevance of the obtained results for the quantum chaos is pointed out.

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On the incredible infinite