Infinite cyclotomic products

TheoremConstancy of an infinite cyclotomic product#●ready to prove

For \(n \ge 1\) and complex \(z\), let \(\hat\Phi_n(z) = \prod_\omega (1 - z\omega)\), the product over the primitive \(n\)-th roots of unity \(\omega\); thus \(\hat\Phi_1(z) = 1 - z\) and \(\hat\Phi_n = \Phi_n\) for \(n \ge 2\), where \(\Phi_n\) is the \(n\)-th cyclotomic polynomial. For every complex \(z\) with \(|z| < 1\), the partial products \(\prod_{n=1}^{N} \hat\Phi_n(z)^{-1/n}\) tend to \(1\) as \(N \to \infty\), where \(\hat\Phi_n(z)^{-1/n} = \exp\bigl(-\tfrac{1}{n} \sum_\omega \log(1 - z\omega)\bigr)\) and \(\log\) is the principal branch of the logarithm.

Each factor \(1 - z\omega\) has positive real part, and this branch equals \(1\) at \(z = 0\), as in the paper's logarithmic-derivative proof. The paper writes the product as \(P(z) = -\prod_{n \ge 1} \Phi_n(z)^{-1/n}\); the sign absorbs \(\Phi_1(z) = -(1 - z)\), and the claim is \(P(z) \equiv 1\).

public theorem_wanted tendsto_prod_cyclotomicHat_inv_root_one {z : ℂ} (hz : ‖z‖ < 1) :
    Tendsto (fun N : ℕ => ∏ n ∈ Finset.Icc 1 N,
        Complex.exp (-(n : ℂ)⁻¹ * ∑ ζ ∈ primitiveRoots n ℂ, Complex.log (1 - z * ζ)))
      atTop (𝓝 1)

Sources

• Hartosh Singh Bal, Constancy of an Infinite Cyclotomic Product via Ramanujan Sums, INTEGERS 25 (2025), Article A96, Theorem 1.

TheoremA zeta-twisted cyclotomic product identity#●ready to prove

For \(n \ge 1\) and complex \(z\), let \(\hat\Phi_n(z) = \prod_\omega (1 - z\omega)\), the product over the primitive \(n\)-th roots of unity \(\omega\). Let \(z\) be a complex number with \(|z| < 1\), let \(s > 1\) be real, and let \(\zeta\) be the Riemann zeta function. Then

\[ \prod_{i \ge 1} (1 - z^i)^{-1/i^s} = \prod_{i \ge 1} \hat\Phi_i(z)^{-\zeta(s)/i^s} \]

in the sense that, as \(N \to \infty\), the partial products over \(1 \le i \le N\) of both sides converge to a common limit. Here \((1 - z^i)^{-1/i^s} = \exp(-i^{-s} \log(1 - z^i))\) and \(\hat\Phi_i(z)^{-\zeta(s)/i^s} = \exp(-\zeta(s) \, i^{-s} \sum_\omega \log(1 - z\omega))\), where \(\log\) is the principal branch of the logarithm.

The exponent \(-\zeta(s)/i^s\) follows arXiv:2511.16975, Theorem 2.2. The INTEGERS record shows \(-\zeta(s)/i\); with that exponent the right side would equal \(1\) by the constancy of \(\prod_{n \ge 1} \hat\Phi_n(z)^{-1/n}\), while the left side does not.

public theorem_wanted exists_tendsto_prod_one_sub_pow_and_cyclotomicHat_zeta {z : ℂ}
    (hz : ‖z‖ < 1) {s : ℝ} (hs : 1 < s) :
    ∃ L : ℂ,
      Tendsto (fun N : ℕ => ∏ i ∈ Finset.Icc 1 N,
          Complex.exp (-(((i : ℝ) ^ s : ℝ) : ℂ)⁻¹ * Complex.log (1 - z ^ i))) atTop (𝓝 L) ∧
      Tendsto (fun N : ℕ => ∏ i ∈ Finset.Icc 1 N,
          Complex.exp (-(riemannZeta (s : ℂ) / (((i : ℝ) ^ s : ℝ) : ℂ)) *
            ∑ ζ ∈ primitiveRoots i ℂ, Complex.log (1 - z * ζ))) atTop (𝓝 L)

Sources

• Hartosh Singh Bal, Constancy of an Infinite Cyclotomic Product via Ramanujan Sums, INTEGERS 25 (2025), Article A96, Theorem 2.

2 ready to prove

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