Cyclotomic polynomials with prescribed height and prime number theory

Abstract

Given any positive integer n, let A(n) denote the height of the n th cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that A(n) is unbounded. We conjecture that every natural number can arise as value of A(n) and prove this assuming that for every pair of consecutive primes p ≥ 127 and q we have q − p ≤ √ p − 1. Using a result of Heath-Brown we show unconditionally that every integer m ≤ x occurs as A(n) value with at most O ǫ (x 3/5+ǫ ) exceptions. On the Lindelöf Hypothesis we show there are at most O ǫ (x 1/2+ǫ ) exceptions. Finally, we study these exceptions further by using deep work of Bombieri-Friedlander-Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier.

Maximum gap in cyclotomic polynomials

“…The investigation on height (maximum absolute value of coefficients) was initiated by the finding that the height can be bigger than 1 (as exhibited by n = 105). It has produced numerous results, to list a few: upper bound [11,10,6,12,36,17,32,7,9,13,31], realizability [37,29,18,19,23,21,22,35,26], and flatness [8,16,27,28,39]. The investigation on jumps (a jump happens when two consecutive coefficients are different) was initiated by Bzdȩga [14] which was motivated by the work of Gallot and Moree [21].…”