A problem about prime numbers and the random walk II
P. Erdös · Illinois Journal of Mathematics · 1961
I am going to prove , 1. Denote by u(a, b) the probability of the random walk passing through a if it starts at b.It is known and easy to prove that (1) u(a, b) cl b a 1-1 (see K. Iw6 AND H. P. McKEAN, JR., Potentials and the random walk, IllinoisJ. Math., vol. 4 (1960), pp.119-132; also a paper of Murdoch cited therein where a sharper result is obtained).In the sequel, the letters p and q de- note primes and u(p, q) u(a, b) in case a (p, 0, 0) and b (q, 0, 0).Consider the number e(n) of points (p, 0, 0) (p -< n) that the path hits.We have to prove that for almost all paths e(n) as n " .