# Dedekind sums by Hans Rademacher, Emil Grosswald.

By Hans Rademacher, Emil Grosswald.

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F o r h = — 1 we have k - l , f c - 2 , - - , l , so t h a t i(-l,fc) = 1 + 2 + - + ( f c - 2 ) F o r h =2, 7(2, fc) = _ = ( f c - >( 2 2 f e - ) 1 (45) becomes 2 , 4 , - - - , f c - l , 1,3,5,•·-,fc — 2 , and + ( _ - lj + ( — - 2j + . · + 1 fc - 1 2 T h e already mentioned particular cases h = — 1 and h = 2 of Zolotareff's Theorem (49) yield the well-known "supplem e n t a r y " theorems of quadratic reciprocity: = ( - ! ) < * - » ' * , and = C-0 < f c I - l V 8 . Finally, if also /i is positive and odd, we obtain from (48) and (4), Sec.

Therefore, Β = Σ J(£)) + Σ £ ( ( £ ) ) + Σ ^ ((£)). l,((i fc))+2 + By Lemma 1, the sum equals ((bc-xja)), Σ *((£)) = so that Σ = The second equality follows directly from (1) (see Chapter 2, B). Handling t h e other two sums in the same way we obtain Β = s(bc,a) + s(ca,b) + s(ab,c). z Σ tmoaabc ((-j-)) \ \ a D . . 1 = ,(l fcc)- l f l 2 / / C i + _ 1 abc + by Lemma 2. Substituting these values of A, B, and C, in (53) we have LATTICE POINTS Sec. Ε 2JV 3 = ^ ( a b c + bc + ca+ab 43 + a + b + c 1 /be ca - ( ν + τ + 3 ~ 2{s(bc,a) + s(ca,b) + ab\\ τ ) ) - + 4 s(ab,c)\ 1 4 1 6 abc abc 12 I 4 1 4 , = 3-abc + J i t o + ca + a b ) + i ( a + b + c) 1 /be ca ab\ .

And = C-0 < f c I - l V 8 . Finally, if also /i is positive and odd, we obtain from (48) and (4), Sec. Ε hl(h,k) 39 LATTICE POINTS + kl(k,h) = -3hk(s(h,k) + + i(h(k-D(k-2) s(k,h)) + k(h-l)(h-2)) + ±(ftfc + h k - 6hk + 2h + 2k) 2 = \(hk 2 2 + h k-3hk-h -k +2h 2 2 2 + 2k-l), or (50) hl(h,k)-rkl(k,h) = i(h-i)(k-\)(h + k-1), a formula of Salie ([32], p . 163). This, of course, together with (49) once more implies immediately the quadratic reciprocity law of the Jacobi symbol E. Lattice Points.