| On products of ratios of consecutive integers (2005) | |||||||||||||||
Abstract | |||||||||||||||
| Products ratios consecutive integers gis Bret che Carl Pomerance rald Tenenbaum For Jean Louis Nicolas his sixtieth birthday Introduction Let a nite sequence with each and write b where the fraction its smallest terms Now N the maximal value N runs through all possible sequences and let denote the corresponding value Note that maximizing instead would lead A obviously have N hence log N log for all shown elegant near tiling the integers with triples that log log log Further brief argument Langevin presented that log Our aim this article establish the true order magnitude for log Put c log log c u max c Theorem For large have log K log Let denote the largest prime factor positive integer with the convention that The lower bound easy consequence the estimate stated the following result Theorem For let denote the number those integers not exceeding such that min c Then for any xed and uniformly for x have x log x de Breteche Pomerance Tenenbaum Remark Under suitably strong form the Elliott Halberstam hypothesis get the better bound c log for This would yield the value Theorem See Section for further methodological remarks The bound probably optimal fact likely the case that c x for where the Dickman Bruijn function even stronger statement suggested Note that follows from selecting n and P n and P and all other cases Indeed with these choices for obtain that for each prime the exponent the prime factorization the rational number N n p n x n Thus log n N log n P c log min We have | |||||||||||||||
Details der Publikation | |||||||||||||||
| |||||||||||||||