Worked solution: Tao's proof of the Erdős discrepancy problem via logarithmically averaged correlations (2015)

Step 2 of 9: Why Dirichlet characters are the test case that almost breaks the conjecture

In plain words

The non-principal character $\chi_3$ modulo $3$ is periodic, completely multiplicative, and has bounded partial sums. It takes the values $0,\pm1$ , so it is not yet an admissible $\{-1,+1\}$ sequence. Tao’s Example 1.4 replaces only the value at the bad prime $3$ by $+1$ , producing the completely multiplicative $\tilde\chi_3$ .

\chi_3 \text{ non-principal Dirichlet character modulo }3 \implies \left|\sum_{j=1}^n \chi_3(jd)\right| \le 3 \text{ for all } n,d

Detailed analysis

For the specific character $\chi_3$ , $\chi_3(n)=0$ exactly when $3\mid n$ , so its partial sums are bounded, while changing the prime value at $3$ to $+1$ gives $\tilde\chi_3$ . For $n=1+3+3^2+\cdots+3^k$ , writing each $j=3^i m$ shows $\sum_{j=1}^n\tilde\chi_3(j)=k+1\gg\log n$ : this is unbounded, but only logarithmically. The proof must rule out this concrete near-counterexample and its twisted variants.

Terms in this step

Dirichlet character: A periodic, completely multiplicative function $\chi$ with period $q$ that vanishes on integers sharing a factor with $q$ and otherwise takes values that are roots of unity; the simplest non-trivial examples take only the values $-1,0,+1$ .
Completely multiplicative function: A function $g:\mathbb{N}\to\mathbb{C}$ satisfying $g(nm)=g(n)g(m)$ for every pair $n,m$ (not just coprime pairs); its value at any integer is determined entirely by its values at primes.

Knowledge used in this step

Szemerédi's theorem