On a generalization of Polya inequality and some of its statistical implications

Let p be a prime and χ a nonprincipal character mod p. Let 1⩽m⩽p and l an integer so that p∤l. Then, we have |∑a=0m−1 χ(a)χ(a+l)|⩽3p ln p. The proof makes use of estimation on Kloosterman sum via the Riemann hypothesis on finite fields. As simple consequence of the theorem we obtain a uniform distribution of consecutive quadratic residues mod p.