On the distribution of modular inverses from short intervals

Abstract

For a prime number $p$ and integer $x$ with $\gcd (x,p)=1$ let $\overline{x}$ denote the multiplicative inverse of $x$ modulo $p.$ In the present paper we are interested in the problem of distribution modulo $p$ of the sequence $$\overline{x},x=1,\dots ,N,$$ and in lower bound estimates for the corresponding exponential sums. As representative examples, we state the following two consequences of the main results. For any fixed $A>1$ and for any sufficiently large integer $N$ there exists a prime number $p$ with $$(\log p{)}^A\asymp N$$ such that $$\underset{\gcd (a,p)=1}{\max }\left|\sum _{x\le N}{\mathbf{e}}_p(a\overline{x})\right|\gg N.$$ For any fixed positive $\gamma <1$ there exists a positive constant $c$ such that the following holds: for any sufficiently large integer $N$ there is a prime number $p>N$ such that $$N>\exp \left(c(\log p\log \log p{)}^{\gamma /(1+\gamma )}\right)$$ and $$\underset{\gcd (a,p)=1}{\max }\left|\sum _{x\le N}{\mathbf{e}}_p(a\overline{x})\right|\gg N^{1-\gamma }.$$