1.3 Coprimality becomes an integer signal
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Let
The Ramanujan sum is
It adds the primitive \(q\)th-root phases selected by QA01. Its definition is complex, but its value is an integer. That integrality is the step that makes the sample-coordinate projector rational. Ramanujan sums and their divisor formulas are classical arithmetic constructions [1].
Theorem QA04: Möbius formula and integrality
Let \(\mu\) be the Möbius function, with \(\mu(1)=1\). For every positive \(q\) and every integer \(n\),
In particular \(c_q\) has period \(q\), is real-valued, satisfies \(c_q(-n)=c_q(n)\), and has \(c_q(0)=\varphi(q)\).
Proof
For a positive integer \(g\),
To see this, only squarefree divisors contribute. If \(g\) has \(r\) distinct prime factors, the sum is \((1-1)^r\), with value one when \(r=0\). Therefore
Substitute this into the phase sum over all \(a\in I_q\), exchange the finite sums, and write \(a=db\) in the terms where \(d\mid a\):
The second equality is QA02. Replacing \(q/d\) by \(e\) gives the displayed divisor sum. It is an integer. Periodicity follows already from the phase definition. The divisor formula is unchanged by \(n\mapsto -n\), so \(c_q(-n)=c_q(n)\). At \(n=0\), every selected phase is one, giving \(c_q(0)=|U_q|\). \(\square\)
At \(q=1\), \(U_1=\{0\}\) because \(\gcd(0,1)=1\). The phase sum and the divisor formula both give \(c_1(n)=1\). This boundary case is included in the proofs even though the inherited arithmetic regression range begins at modulus two.
Theorem QA05: Exact spectral support and energy
Over one period,
Moreover,
Proof
Substitute the phase definition and exchange two finite sums:
QA02 makes the inner sum \(q\) exactly when \(a=k\), and zero otherwise. The energy identity follows from QA03:
\(\square\)
The selector has been recovered by an identity. A numerical spectrum can illustrate the result, but neither a tolerance nor a peak-finding rule defines the arithmetic support.