Pretentious Analytic Number Theory and Multiplicative Functions

The "pretentious" approach, developed by Granville-Soundararajan, studies multiplicative functions by measuring their distance from characters. This framework provides a unified perspective on many classical results without directly using zeros of L-functions.

Pretentious Distance

Definition6.6Pretentious Distance

For multiplicative functions $f, g: \mathbb{N} \to \mathbb{C}$ with $|f|, |g| \leq 1$ , the pretentious distance is: $\mathbb{D}(f, g; x)^2 = \sum_{p \leq x} \frac{1 - \mathrm{Re}(f(p)\overline{g(p)})}{p}$ This is a pseudometric on the space of multiplicative functions. The key philosophy: $f$ "pretends to be" $g$ if $\mathbb{D}(f, g; x)$ is bounded as $x \to \infty$ .

ExampleHalász's Theorem via Pretentious Methods

Halász's theorem (1968, reformulated pretentiously): Let $f$ be a multiplicative function with $|f| \leq 1$ . Then $\frac{1}{x}\sum_{n \leq x} f(n) \to 0$ unless $f$ "pretends to be" $n^{it}$ for some $t \in \mathbb{R}$ , i.e., $\min_t \mathbb{D}(f, n^{it}; x) < \infty$ . Quantitatively: $\left|\sum_{n \leq x} f(n)\right| \ll x \exp(-c \min_{|t| \leq T} \mathbb{D}(f, n^{it}; x)^2)$ for suitable $T$ .

Mean Values of Multiplicative Functions

Definition6.7Mean Value Theorems

The study of $M(f, x) = \sum_{n \leq x} f(n)$ for multiplicative $f$ connects to the Dirichlet series $F(s) = \sum f(n)n^{-s}$ . Key results:

Wirsing's theorem: If $f$ is real non-negative multiplicative with $\sum f(p)/p$ divergent, then $M(f,x) \sim cx/\log^{1-\alpha}x$ for a computable $\alpha$ .
Halász's theorem: Provides the asymptotic of $M(f,x)$ for complex-valued $f$ , showing oscillation governed by the "pretentious target" $n^{it}$ .
Granville-Soundararajan: Unified these and Matomäki-Radziwiłł (mean values in short intervals) via the pretentious framework.

RemarkMatomäki-Radziwiłł Theorem

The breakthrough theorem of Matomäki-Radziwiłł (2015): for any multiplicative $f$ with $|f| \leq 1$ , $\frac{1}{H}\sum_{x < n \leq x+H} f(n)$ equals its expected average for "almost all" $x \in [X, 2X]$ , as long as $H \to \infty$ (arbitrarily slowly). This implies the Möbius function has cancellation in almost all short intervals $[x, x+H]$ with $H \to \infty$ , a result previously out of reach.

Applications

ExampleWhen Does $f$ Pretend to Be a Character?

If $\mathbb{D}(f, \chi; x) < \infty$ for a Dirichlet character $\chi$ mod $q$ , then $f(n)$ is distributed like $\chi(n)$ on average. The pretentious approach recovers the PNT: since $\mu(n)$ does not pretend to be any $n^{it}$ , Halász gives $\sum_{n \leq x} \mu(n) = o(x)$ , which is equivalent to $\zeta(1+it) \neq 0$ .