Alternative Methods for Evaluating r s (n)

Emil Grosswald
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Emil Grosswald: Temple University, College of Liberal Arts

Chapter Chapter 13 in Representations of Integers as Sums of Squares, 1985, pp 175-187 from Springer

Abstract: Abstract Theorem 12.1 is formulated in [72] as follows: Theorem 1. For integers h, k with (h, k) = 1, let $$ \lambda \left( {h/k} \right) = {\left( {2k} \right)^{{ - 1}}}\sum\nolimits_{{q = 1}}^{{2k}} {{e^{{2\pi ih{q^{2}}/2k}}}} $$ and set $$ {A_{k}} = \sum\nolimits_{{1 \le h \le 2k,\left( {h,k} \right) = 1}} {{\lambda ^{s}}{e^{{ - 2\pi ihn/2k}}}} $$ . Then, if $$ S\left( n \right) = \sum\nolimits_{{k = 1}}^{\infty } {{A_{k}}} $$ and s = 5, 6, 7, or 8, one has for some constant c = c(s), independent of n, that $${r_{s}}\left( n \right) = c\left( s \right){n^{{\left( {s/2} \right) - 1}}}S\left( n \right) $$ .

Date: 1985
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DOI: 10.1007/978-1-4613-8566-0_14

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