arXiv subjects
Publications and source records attributed to Christian Elsholtz.
We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.
It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one subsequence). When the length L of the sequence is much longer, we might expect that there are 2^L/N subsequences with this property (imagine the subsequences have sum-of-terms uniformly distributed modulo N -- the 0 class gets about 2^L/N subsequences); however, it is easy to see that this is actually false. Nonetheless, we are able to prove that if the initial sequence has length at least 4N, and N is odd, then there is a subsequence of length L > N, having at least 2^L/N subsequences that sum to 0 mod N.
Suppose that P is an infinite set of primes such that P = A + B + C, where A,B,C are sets with at least two elements. We show that if P(x) > c x/log^d x (where P(x) = the number of elements of P that are <= x), and if A,B,C is a "regular" triple of sets, then either |A+B| <= d, or |B+C| <= d, or |A+C| <= d.