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I ran into two problems involving sums of Legendre symbols: for $p$ prime and $a,b,k$ integers:

$$\sum_{x=1}^{p} \left(\frac{x(x+k)}{p}\right)$$$$ \sum_{x=1}^{p} \left(\frac{x^2 +ax + b}{p}\right)$$

How can I evaluate these? I have some basic background working with Legendre symbols but not with sums of series.

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For the first sum: If $k\equiv 0\mod p$ the first sum is $p-1$. Hence if we assume $k\not \equiv 0 \mod p$ $$\sum_{x \mod p}\bigg(\frac{x^2+kx}{p}\bigg)=\sum_{x \neq 0\mod p}\bigg(\frac{x^2+kx}{p}\bigg)\\ =\sum_{x\neq 0 \mod p}\bigg(\frac{x^2}{p}\bigg)\bigg(\frac{1+kx^{-1}}{p}\bigg)$$ as $\big(\frac{x^2}{p}\big)=1$ for $x\neq 0$ we have $$\sum_{x \mod p}\bigg(\frac{x^2+kx}{p}\bigg)=\sum_{x\neq 0 \mod p}\bigg(\frac{1+kx^{-1}}{p}\bigg)$$ As $\{x :x\neq 0 \mod p\}=\{kx^{-1}: x\neq 0 \mod p\}$ we have
$$ \sum_{x\neq 0 \mod p}\bigg(\frac{1+kx^{-1}}{p}\bigg)=\sum_{x\neq 0 \mod p}\bigg(\frac{1+x}{p}\bigg)\\ =\sum_{x\mod p}\bigg(\frac{x}{p}\bigg)-\bigg(\frac{1}{p}\bigg)\\ =-1.$$ Hence the first sum is $p-1$ if $k\equiv 0 \mod p$ and $-1$ if $k\not \equiv 0 \mod p$. I have an explicit expression for the for the second sum in terms of $(\frac{b-2^{-2}a^2}{p})$ and $(\frac{-1}{p})$ but the proof is too long to write it here.

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  • $\begingroup$ Could you clarify the step why $\{x :x\neq 0 \mod p\}=\{kx^{-1}: x\neq 0 \mod p\}$? $\endgroup$ – F. Yan May 13 '17 at 3:38
  • $\begingroup$ The set of non zero residues modulo $p$ is a multiplicative group and if $k,x\neq 0$ then clearly $kx^{-1}\neq 0$. Hence $\{kx^{-1}:x\neq 0 \mod p\}$ is a subset of $\{x:x\neq 0\mod p\}$. Every $y\in \{x:x\neq 0\mod p\}$ belongs to $\{kx^{-1}:x\neq 0 \mod p\}$ as when $x=ky^{-1}$, $kx^{-1}=y$. Hence $\{x:x\neq 0\mod p\}$ is a subset of $\{kx^{-1}:x\neq 0 \mod p\}$ which implies $\{x:x\neq 0\mod p\}=\{kx^{-1}:x\neq 0 \mod p\}.$ $\endgroup$ – Sai Teja May 13 '17 at 4:02

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