Patterns and complexity of multiplicative functions

Abstract

I. Schur and G. Schur proved that, for all completely multiplicative functions f : N → {−1, 1}, with the exception of two character-like functions, there is always a solution of f (n) = f (n + 1) = f (n + 2) = 1. Hildebrand proved that for the Liouville λ-function each of the eight possible sign combinations (λ(n), λ(n + 1), λ(n + 2)) occurs infinitely often. We prove for completely multiplicative functions f : N → {−1, 1}, satisfying certain necessary conditions, that any sign pattern ( 1, 2, 3, 4), i ∈ {−1, 1}, occurs for infinitely many 4-term arithmetic progressions (f (n), f(n + d), f(n + 2d), f(n + 3d)).The proof introduces graph theory and new combinatorial methods to the subject.

Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations

“…As a partial result in this direction Buttkewitz and Elsholtz [2] proved for a completely multiplicative function f : N → {−1, 1}, with at least two primes p 1 , p 2 such that f (p 1 ) = f (p 2 ) = −1, and assuming that f = f ± , where…”

The Erdős discrepancy problem

“…Thus f takes values in the unit sphere of H. Repeating the calculation in Example 1. 4, we see that if n = 1 + 3 + 3 2 + · · · + 3 k , then log n.…”

“…This reduction is obtained by a standard compactness argument 4 , but we give the details for sake of completeness. Suppose that for each X, we have such a g X obeying (2.2) as above.…”

“…This can be compared with random sequences f : N → {−1, +1}, whose discrepancy would be expected to diverge like N 1/2+o (1) . See the paper of Borwein, Choi, and Coons [2] for further analysis of functions such asχ 3 , which seems to have been first discussed in [20]; see also [4] for some further discussion of the sign patterns inχ 3 . One can also reduce the discrepancy of this example slightly (by a factor of about two) by changing the value of the completely multiplicative functionχ 3 at 3 from +1 to −1.…”

Zero-Sum Squares in Bounded Discrepancy $\{-1,1\}$-Matrices

“…Caro et al proved in [6] that for any finite sequence f : [1, n] → {−1, 1} satisfying that | n i=1 f (i)| is small, there is a set of consecutive numbers B ⊂ [1, n] for which i∈B f (i) is also small (in particular, small can mean zero-sum). Another interesting work is [4], where Buttkewitz and Elsholtz proved the existence of zero-sum arithmetic progressions with four terms in certain sequences f : N → {−1, 1}. Balister et al studied matrices where, for some fixed integer p, the sum of each row and each column is a multiple of p [3]; they showed that these matrices appear in any large enough integer square matrix.…”