By Cassels, John William Scott
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Nevertheless, they are not just a gratuitous generalization since they enable us, in particular, to deal with virtual geometries and their associated spectra— suitably deﬁned by means of their zeta functions—as though they arose from actual fractal geometries. In Chapters 9, 10 and 11, the extra ﬂexibility of this framework allows us to study the zeros of several classes of zeta functions. The original explicit formula was given by Riemann [Rie1] in 1858 as an analytical tool to understand the distribution of primes.
32) where k, j = 1, 2, 3, . . ; that is, they are the integer multiples of the reciprocal lengths of L. The total multiplicity of the frequency f is equal to (ν) wf = wl = w1/f + w2/f + w3/f + · · · . 33) l: f ·l∈N∗ To study the frequencies, we introduce the spectral counting function and the spectral zeta function. Note that Nν and ζν depend on L. However, for simplicity, we do not indicate this explicitly in our notation. 20. 35) f which converges for Re s suﬃciently large. 35), the sum is extended over all distinct frequencies of L.
150]. 1, and the notes to Chapter 9. The primary motivations of the authors of those papers came from the investigation of the oscillatory phenomena in the geometry and the spectrum of fractal drums [Lap3] (including self-similar drums) and, in particular, of fractal strings, where the connections between direct or inverse spectral problems and the Riemann zeta function or the Riemann hypothesis were ﬁrst discovered in [LapPo1, 2] or [LapMa1, 2], respectively. 7 for a sample of references in the physics literature—of which we have become aware recently and with rather diﬀerent motivations, coming from the study of turbulence, lacunarity, biophysics, and other applications.
Rational quadratic forms by Cassels, John William Scott