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The Dirichlet series Lm(s) are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with the Dirichlet series, denoted by {sm,n}n≥0. We obtain a formula for the exponential generating function sm(x) of sm,n, where m is an arbitrary positive integer. In particular, for m > 1, say, m = bu2, where b is square-free and u > 1, we show that sm(x) can be expressed as a linear combination of the four functions w(b, t) sec(btx)(±cos((b - p)tx) ± sin(ptx)), where p is a nonnegative integer not exceeding b, t|u2 and w(b, t) = Kbt/u with Kb being a constant depending on b. Moreover, the Dirichlet series Lm(s) can be easily computed from the generating function formula for sm(x). Finally, we show that the main ingredient in the formula for sm,n has a combinatorial interpretation in terms of the ?-alternating augmented m-signed permutations defined by Ehrenborg and Readdy. More precisely, when m is square-free, this answers a question posed by Shanks concerning a combinatorial interpretation of the numbers sm,n. When m is not square-free, say m = bu2, the numbers K-1 b sm,n can be written as a linear combination of the numbers of ?-alternating augmented bt-signed permutations with integer coefficients, where t|u2. [ABSTRACT FROM AUTHOR]

We report optical emission of SiC nanocrystallite films, which clearly shows the quantum confinement effect. Bulk polycrystalline 3C-SiC was first electrochemically etched and then the fabricated porous silicon carbide was ultrasonically treated in water or toluene suspension to disperse into colloidal nanoparticles. Transmission electron microscopy images clearly show that the colloidal nanoparticles have 3C-SiC lattice structure with sizes varying from about 6 nm down to below 1 nm. The suspension of 3C-SiC nanocrystallites exhibits ultrabright emission with wavelengths ranging from 400 to 520 nm when the excitation wavelength varies from 250 to 480 nm, in accordance with the quantum confinement effect. By adding polystyrene to the toluene suspension containing SiC nanoparticles and coating the mixing solution onto a Si wafer, we obtain the SiC/polystyrene films that luminesce. [ABSTRACT FROM AUTHOR]