How do I prove that the following quantity is purely imaginary:
$$\sum_{0\leq l_1<l_2<l_3<l_4\leq q-1} e^{-2\pi i \frac{(l_1^2-l_2^2+l_3^2-l_4^2)}{q} } $$ where $q$ is an odd number?
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SK1712
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Here $q$ "is" an odd number. No assumtions. The above quantity is imaginary for $q$ odd, which can be verified using a machine but I want to prove it analytically. – SK1712 Sep 14 '17 at 10:42
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Maybe try to show it is invariant under sending $a+bi$ to $-a+bi$. – M. Van Sep 14 '17 at 10:42
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3or better: try to show that complex conjugation results in multiplying by $-1$! – M. Van Sep 14 '17 at 10:44
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@M.Van Yeah, I have tried to sum up the quantity and its conjugate and simplify the expression. But, couldn't succeed. – SK1712 Sep 14 '17 at 10:51
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What is the source of this problem? – lhf Sep 14 '17 at 12:11
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or try squaring it – M. Van Sep 14 '17 at 12:48
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It seems to be $0$ for $q$ prime. – lhf Sep 14 '17 at 13:12
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@lhf That could be. Actually, it's part of a problem where I need to compare the real parts of two terms let's say A and B=B'+B'', so we've real(A) = real(B') + real(B''). Here, if B'' is purely imaginary or zero then we've real(A) = real(B') which is desired. In this case, B'' is term mentioned above. – SK1712 Sep 14 '17 at 15:21
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That made me feel old. Any way to enlarge that summation without also enlarging all the other text on this page? – David R. Sep 20 '17 at 19:43