Binomial Coefficients and the Distribution of the Primes
| dc.creator | Xylouris, Triantafyllos | |
| dc.date | 2007-09-28 | |
| dc.date.accessioned | 2026-07-07T08:32:57Z | |
| dc.date.available | 2026-07-07T08:32:57Z | |
| dc.description | Let omega(n) be the number of distinct prime factors dividing n and m > n natural numbers. We calculate a formula showing which prime numbers in which intervals divide a given binomial coefficient. From this formula we get an identity omega(binom(nk)(mk))=sum_i (pi(k/b(i))- pi(k/a(i))) + O(sqrt(k)). Erdoes mentioned that omega(binom(nk)(mk))= log n^n/(m^m (n-m)^(n-m)) k/log k + o(k/log k). As an application of the above identities, we conclude some well-known facts about the distribution of the primes and deduce for all natural numbers k an expression (also well-known) log k = sum_i a_k(i) which generalizes log 2 = sum_i^(infty) (-1)^(j+1) / j. | |
| dc.description | 15 pages | |
| dc.identifier | https://arxiv.org/abs/0709.4676 | |
| dc.identifier | http://arxiv.org/abs/0709.4676 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/138961 | |
| dc.subject | Number Theory | |
| dc.subject | 11A41;11B65 | |
| dc.title | Binomial Coefficients and the Distribution of the Primes | |
| dc.type | text |