# Can a upper bound of $\sum_{b=1}^{p-1}\left(\frac{b^2-a^2}{p}\right)\left(\frac{b^2-1}{p}\right)$, $a\in{Z}$, be strictly less than $\sqrt{p}$?

By weil estimate I can only say, one bound is $$\sqrt{p}$$. Can it be strictly less than $$\sqrt{p}$$? I want to see whether one better bound be given or not.

• Your question should be clear without the title. After the title has drawn someone's attention to the question by giving a good description, its purpose is done. The title is not the first sentence of your question, so make sure that the question body does not rely on specific information in the title. – Martin R Jul 12 at 7:22
• Thanks. I have made necessary changes. – Nilanjan Bag Jul 12 at 9:17
• I think @MartinR means you should phrase your question (completely) in the question body, so that the question is understandable without the title. As of now, without the title, there is no mention of what is to be bounded. – awllower Jul 13 at 3:00