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 Incoming links: Large Character Sums: Pretentious Characters and the Pólya-Vinogradov Theorem by Andrew Granville and K. Soundararajan

Triangle inequality for pretentious distance Created 2026-09-24 Updated 2026-11-05

For multiplicative functions

f_{1}, f_{2}, g_{1}, g_{2}

bounded in modulus by one, the pretentious distance satisfies

D (f_{1} f_{2}, g_{1} g_{2}; x) \leq D (f_{1}, g_{1}; x) + D (f_{2}, g_{2}; x) .

Here products mean pointwise products of arithmetic functions. For complex numbers

z, w

of modulus at most one, write

a = ℜ z

and

b = ℜ w

. The bounds

∣ℑ z ∣ \leq 2 (1 - a)

and

∣ℑ w ∣ \leq 2 (1 - b)

give

1 - ℜ (z w) = (1 - a) + (1 - b) - (1 - a) (1 - b) + ℑ z ℑ w \leq (1 - a + 1 - b)^{2} .

Apply this with

z = f_{1} (p) \overline{g_{1} (p)}

and

w = f_{2} (p) \overline{g_{2} (p)}

at each prime

p \leq x

, divide by

p

, and use the triangle inequality for the resulting finite Euclidean norm. This proves the displayed product inequality.^[ref]

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