A Treatise On the Theory of Functions by James. Harkness

By James. Harkness

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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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