  We confirm the eventual evasiveness of several classes of monotone graph
properties under widely accepted number theoretic hypotheses. In particular we
show that Chowla's conjecture on Dirichlet primes implies that (a) for any
graph $H$, "forbidden subgraph $H$" is eventually evasive and (b) all
nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are
eventually evasive. ($n$ is the number of vertices.)
  While Chowla's conjecture is not known to follow from the Extended Riemann
Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's $L$ functions), we show
(b) with the bound $O(n^{5/4-\epsilon})$ under ERH.
  We also prove unconditional results: (a$'$) for any graph $H$, the query
complexity of "forbidden subgraph $H$" is $\binom{n}{2} - O(1)$; (b$'$) for
some constant $c>0$, all nontrivial monotone properties of graphs with $\le
cn\log n+O(1)$ edges are eventually evasive.
  Even these weaker, unconditional results rely on deep results from number
theory such as Vinogradov's theorem on the Goldbach conjecture.
  Our technical contribution consists in connecting the topological framework
of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti,
Khot, and Shi (2002), with a deeper analysis of the orbital structure of
permutation groups and their connection to the distribution of prime numbers.
Our unconditional results include stronger versions and generalizations of some
result of Chakrabarti et al.
