MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a result which gives a lower bound on the number of inverse pairs $(a, a^{-1})$ modulo prime $p$ lying in the interval $[1,t]$, where $t < p$?

share|cite|improve this question
The expected number of such pairs is about $t^2/p$. That the actual number is $t^2/p + O(p^{1/2} \log^2 p)$ follows from the upper bound $2 p^{1/2}$ on the size of a Kloosterman sum. In particular there's a nontrivial lower bound for $t \gg p^{3/4} \log p$. $$ $$ This bound (also for length-$t$ intervals mod $p$ other than $[1,t]$) appears as a Lemma in Merel's proof that for each $d$ there's an (explicit though large) upper bound on the size of the torsion group of an elliptic curve over any number field of degree $d$. – Noam D. Elkies Jan 7 '13 at 1:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.