- #1
Math100
- 802
- 222
- Homework Statement
- For each real-valued nonprincipal character ## \chi\pmod {16} ##, verify that ## A(225)\geq 1 ##.
- Relevant Equations
- Let ## \chi ## be any real-valued character mod ## k ## and let ## A(n)=\sum_{d\mid n}\chi(d) ##. Then ## A(n)\geq 0 ## for all ## n ##, and ## A(n)\geq 1 ## if ## n ## is a square.
Let ## n=225 ## and ## d ## be the divisors of ## n ##.
Then ## d=\left \{ 1, 3, 5, 9, 15, 25, 45, 75, 225 \right \} ##.
Note that the real-valued nonprincipal characters ## \chi\pmod {16} ## are ## \chi(1), \chi(7), \chi(9), \chi(15) ##.
Observe that ## \chi(1)=1, \chi(7)=\pm 1, \chi(9)=\pm 1, \chi(15)=\pm 1 ##.
Thus ## A(225)=\sum_{d\mid 225}\chi(d)=\chi(1)+\chi(9)+\chi(15)=1+1+1=3\geq 1 ##.
Therefore, ## A(225)\geq 1 ##.
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline n & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\
\hline \chi_{1}(n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\hline \chi_{2}(n) & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\
\hline \chi_{3}(n) & 1 & i & i & 1 & -1 & -i & -i & -1 \\
\hline \chi_{4}(n) & 1 & -i & i & -1 & -1 & i & -i & 1 \\
\hline \chi_{5}(n) & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\
\hline \chi_{6}(n) & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\
\hline \chi_{7}(n) & 1 & i & -i & -1 & -1 & -i & i & 1 \\
\hline \chi_{8}(n) & 1 & -i & -i & 1 & -1 & i & i & -1 \\
\hline
\end{array}
Then ## d=\left \{ 1, 3, 5, 9, 15, 25, 45, 75, 225 \right \} ##.
Note that the real-valued nonprincipal characters ## \chi\pmod {16} ## are ## \chi(1), \chi(7), \chi(9), \chi(15) ##.
Observe that ## \chi(1)=1, \chi(7)=\pm 1, \chi(9)=\pm 1, \chi(15)=\pm 1 ##.
Thus ## A(225)=\sum_{d\mid 225}\chi(d)=\chi(1)+\chi(9)+\chi(15)=1+1+1=3\geq 1 ##.
Therefore, ## A(225)\geq 1 ##.
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline n & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\
\hline \chi_{1}(n) & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
\hline \chi_{2}(n) & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \\
\hline \chi_{3}(n) & 1 & i & i & 1 & -1 & -i & -i & -1 \\
\hline \chi_{4}(n) & 1 & -i & i & -1 & -1 & i & -i & 1 \\
\hline \chi_{5}(n) & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \\
\hline \chi_{6}(n) & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \\
\hline \chi_{7}(n) & 1 & i & -i & -1 & -1 & -i & i & 1 \\
\hline \chi_{8}(n) & 1 & -i & -i & 1 & -1 & i & i & -1 \\
\hline
\end{array}
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