DENOMINATORS OF BERNOULLI POLYNOMIALS

作者：Bordelles Olivier; Luca Florian; Moree Pieter; Shparlinski Igor E

来源：Mathematika, 2018, 64(2): 519-541.

DOI：10.1112/S0025579318000153

For a positive integer n let (SIM)(n )= Pi(p sp(n)>= p )p(,) where p runs over primes and s(p)(n) is the sum of the base p digits of n. For all n we prove that (SIM)(n) is divisible by all "small" primes with at most one exception. We also show that (SIM)(n )is large and has many prime factors exceeding root n, with the largest one exceeding n(20/37). We establish Kellner's conjecture that the number of prime factors exceeding root n grows asymptotically as k root/logn for some constant tc with k = 2. Further, we compare the sizes of (SIM)(n) and (SIM)(n+1), leading to the somewhat surprising conclusion that although (SIM)(n) tends to infinity with n, the inequality (SIM)(n) > (SIM)(n+1 )is more frequent than its reverse.

• 出版日期2018