Emerging Applications of Number Theory by Pavel Bleher (auth.), Dennis A. Hejhal, Joel Friedman,

By Pavel Bleher (auth.), Dennis A. Hejhal, Joel Friedman, Martin C. Gutzwiller, Andrew M. Odlyzko (eds.)

Most humans are inclined to view quantity idea because the very paradigm of natural arithmetic. With the appearance of desktops, even though, quantity thought has been discovering more and more purposes in functional settings, equivalent to in cryptography, random quantity iteration, coding idea, or even live performance corridor acoustics. but different purposes are nonetheless rising - delivering quantity theorists with a few significant new components of chance. The 1996 IMA summer season software on rising purposes of quantity idea was once geared toward stimulating additional paintings with a few of these latest (and most fascinating) functions. focus was once on quantity theory's contemporary hyperlinks with: (a) wave phenomena in quantum mechanics (more particularly, quantum chaos); and (b) graph concept (especially expander graphs and similar spectral theory). This quantity comprises the contributed papers from that assembly and should be of curiosity to someone intrigued by way of novel purposes of recent number-theoretical techniques.

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5) The functions f m ( t) are periodic functions of period 1. 1 is played by the following lemma. 2 (See [BBl). 2: L Irm(O:W« M~+c, m5,M Irm(O:W > cM~, m 4 the number of solutions of the equation n 21 + ... +nd2 =m is of the order of m(d-2)/2, which gives rise to the jumps of the function No(R). For d = 3 Jarnik [Jar] proves the following result.

2: L Irm(O:W« M~+c, m5,M Irm(O:W > cM~, m 4 the number of solutions of the equation n 21 + ... +nd2 =m is of the order of m(d-2)/2, which gives rise to the jumps of the function No(R). For d = 3 Jarnik [Jar] proves the following result. Let No(R) = #{n E Z3: Inl ~ R}, and Fo(R) = No(R) _ R 41rR3 3 R2:1. , Put Fo(R) = 0 for R < 1. 3 (See [Jar]).

H. HARDY, On Dirichlet's divisor problem, Proc. London Math. Soc. 15, 1-25 (1916). H. HARDY AND E. H. Hardy, Vol. 2, 329-356, Oxford, 1967. R. HEATH-BROWN, The distribution and the moments of the error term in the Dirichlet divisor problem, Acta Arith. 60, 389-415 (1992). [Horl] L. HORMANDER, The spectral function of an elliptic operator, Acta Math. 121, 193-218 (1968). [Hor2] L. HORMANDER, Fourier integral operators I, Acta Math. 127, 79-183 (1971). N. HUXLEY, Exponential sums and lattice points, Proc.

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