Some unsolved problems.

Assumption University of Windsor sponsored a symposium for mathematicians from Ontario, Michigan, and Indiana, The symposium gave occasion for an informal lecture in which I discussed various old and new questions on number theory, geometry and analysis. In the following list, I record these problems, with the addition of references and of a few further questions. A. NUMBER THEORY 1. It is known [35, Vol. 1, Section 581 that n(2x) < 257(x) for sufficiently large x. Is it true that (1) n(x + y) < B(X) + n(y) ?-Ungrir has verified the inequality for y 5 41. Hardy and Littlewood [29, p. 691 have proved that (2) P(X + Y)-n(x) < cy/log y. In the same paper, they discuss many interesting conjectures. They put h-0 sup [R (x + Y)-n(x)1 = p(y) , x-00 and they conjecture that p(y) > y/log y, and that perhaps n(y)-p(y)-oo as y-) 00. Hardy and Littlewood deduce (2) by Bruds method. A very difficult conjecture, weaker than (1) but much stronger than (2), is that corresponding to each s > 0 there exists a yE such that, for y > yE, dx + Y)-dy) < (1 + dy/log y-It has not yet been disproved that p(y) = 1 for all y. If p(y) > 1 for some y, then lim inf (h+l-pn) < 00. 2. About seventy years ago, Piltz [38] conjectured that, for each c > 0, pn+l-pn= O(n&). Cram& conjectured [7, p. 241 that pn+l-pn = O((log n)?. If lim sup (p,+l-p,)/(log n) " = 1 , then, for each E > 0, infinitely many of the intervals [n, n + (1-s)(log n) " ] contain no primes, but for n > ns, there is a prime between n and n + (1 + E)(log n) ". The Riemann hypothesis implies that prl+1-pn < n&+1/2 [35, Vol. 1, p. 3381. Thus the old conjecture that there is always a prime between two consecutive squares already goes beyond the Riemann hypothesis.