2015/11/17 by William D. Banks, Banks, William D., Victor Z. Guo +1
Mathematics · #11L40 #11N37 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11L40 #msc:11N37
paper · pdf · doi:10.48550/arxiv.1511.05523
8 pages
arxiv created 2015/11/17 · arxiv updated 2015/11/18
For any odd prime number p, let (⋅|p) be the Legendre symbol, and let n1(p)<n2(p)<⋯ be the sequence of positive nonresidues modulo p, i.e., (nk|p)=-1 for each k. In 1957, Burgess showed that the upper bound n1(p)≪εp^(4√(e))-1+ε holds for any fixed ε>0. In this paper, we prove that the stronger bound nk(p)≪ p^(4√(e))-1exp(√e-1log ploglog p ) holds for all odd primes p, where the implied constant is absolute, provided that k≤ p^(8√(e))-1 exp(\tfrac12√e-1log ploglog p-\tfrac12loglog p). For fixed ε∈(0,(π-2)/(9π-2)] we also show that there is a number c=c(ε)>0 such that for all odd primes p and either choice of θ∈\± 1\, there are ≫εy/(log y)ε natural numbers n≤ y with (n|p)=θ provided that y≥ p^(4√(e))-1exp(c(log p)1-ε).