On Cilleruelo's conjecture for the least common multiple of polynomial sequences
Heilbronn Number Theory Seminar
23rd January 2019, 4:00 pm – 5:00 pm
Howard House, 4th Floor Seminar Room
It is an elementary exercise that the logarithm of the least common multiple of all integers 1, 2, ..., N is exactly given by the Chebyshev function, the partial sum of the von Mangoldt function from 1 to N. Hence, by the Prime Number Theorem, log(lcm[1, ..., N]) is asymptotically equal to N.
In contrast, a relatively recent conjecture due to Cilleruelo states that for an irreducible nonlinear polynomial f with integer coefficients of degree d > 1, the least common multiple of the sequence f(1), f(2), ..., f(N) has asymptotic growth (d-1)*N*log(N) as N goes to infinity. I will discuss the background and status of this conjecture, and time permitting will discuss recent work with Sa'ar Zehavi, where we establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of N depending on the range of shifts.