On the Erd˝ os Discrepancy Problem*

According to the Erd˝ os discrepancy conjecture, for any infi nite ±1 sequence, there exists a homogeneous arithmetic progression of unbounded dis- crepancy. In other words, for any ±1 sequence (x1,x2,...) and a discrepancy C, there exist integers m and d such that | Pm=1 xi·d| > C. This is an 80-year-old open problem and recent development proved that this conjecture is true for dis- crepancies up to 2. Paul Erd˝ os also conjectured that this property of unbounded discrepancy even holds for the restricted case of completely multiplicative se- quences (CMSs), namely sequences (x1,x2,...) where xa·b = xa · xb for any a,b � 1. The longest CMS with discrepancy 2 has been proven to be of size 246. In this paper, we prove that any completely multiplicative sequence of size 127, 646 or more has discrepancy at least 4, proving the Erd˝ os discrepancy con- jecture for CMSs of discrepancies up to 3. In addition, we prove that this bound is tight and increases the size of the longest known sequence of discrepancy 3 from 17,000 to 127,645. Finally, we provide inductive construction rules as well a s streamlining methods to improve the lower bounds for sequences of higher dis- crepancies.