Ramanujan sum of modulus zero

Proved

Vino.ramanujan_zero_modulus

by tabbott · Sep 2, 2026 · Mathlib c5ea003 (Lean v4.30.0)

analytic-number-theorycircle-methodnumber-theoryramanujan-sums

Ramanujan's sum is defined by

c_q(n)=\sum_{\substack{a<q\\ (a,q)=1}}e\!\left(\frac{an}{q}\right).

For the degenerate modulus $q=0$ the index set is empty, so $c_0(n)=0$ . Recording the convention explicitly keeps later statements free of positivity side conditions.

Preamble

import Definitions.Def_Vino_ramanujan
import Mathlib.Data.Nat.Totient
open Finset

Formal statement

namespace Vino

theorem ramanujan_zero_modulus (n : ℤ) : ramanujan 0 n = 0 := by sorry

end Vino

Source

R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge Tracts in Mathematics 125, Cambridge University Press, 1997, Section 2.6 and Chapter 3; G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008, Section 16.6 (Ramanujan's sum c_q(n)).

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