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Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M. V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl's conjecture for the asymptotics of the eigenvalues of the Laplacian on a bounded open subset of W (see [16]). We solve here in the one-dimensional case (that is, when n = 1) this 'modified Weyl-Berry conjecture'. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. We therefore show that one can 'hear' (that is, recover from the spectrum) not only the Minkowski fractal dimension of the boundary-as was established previously by the first author-but also, under the stronger assumptions of the conjecture, its Minkowski content (a 'fractal' analogue of its 'length').We also prove (still in dimension one) a related conjecture of the first author, as well as its converse, which characterizes the situation when the error estimates of the aforementioned paper are sharp.
Abstract. In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the "number field sieve", which was invented by John Pollard. The present paper is devoted to the description and analysis of a more general version of the number field sieve. It should be possible to use this algorithm to factor arbitrary integers into prime factors, not just integers of a special form like the ninth Fermat number. Under reasonable heuristic assumptions, the analysis predicts that the time needed by the general number field sieve to factor n is exp((c+o(l))(logn)1/3(loglogn)2/3) (for n-*oo), where c=(64/9)l/3 = 1.9223. This is asymptotically faster than all other known factoring algorithms, such äs the quadratic sieve and the elliptic curve method. There does not yet exist an Implementation of the number field sieve for general integers, so that a practical comparison cannot yet be made.
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