bibkey: salem1953integralequation authors: Raphaël Salem year: 1953 title: Sur une proposition équivalente à l’hypothèse de Riemann doi: null url: https://gallica.bnf.fr/ark:/12148/bpt6k3188h claim: Bounded integral-equation uniqueness, the positive logarithmic kernel, and its Mellin–Fourier connection to zeta zeros. strata_touched:

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Salem’s integral equation

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Raphaël Salem, Sur une proposition équivalente à l’hypothèse de Riemann, Comptes rendus hebdomadaires des séances de l’Académie des sciences, volume 236 (1953), printed pages 1127–1128. The digitized volume is at https://gallica.bnf.fr/ark:/12148/bpt6k3188h. The supplied original-page source packet identifies both printed pages and contains selected visual transcriptions. These transcriptions, rather than a new inspection of the images or a complete prose transcription, were read here.

The first paragraph on page 1127 gives a necessary and sufficient condition for no zeta zero of abscissa sigma, with 0 < sigma < 1. Its displayed equation (1) is

[ \int_{-\infty}^{\infty} \frac{e^{-\sigma v}\varphi(v)}{e^{e^{y-v}}+1},dv=0. ]

The ensuing text explicitly assumes sigma > 0. Page 1128 displays

[ \Gamma(s)(1-2^{1-s})\zeta(s) =\int_0^\infty\frac{t^{s-1}}{e^t+1},dt =\int_{\mathbb R}\frac{e^{\sigma u}}{e^{e^u}+1}e^{i\gamma u},du, \qquad s=\sigma+i\gamma. ]

It names the kernel K_sigma(u) = exp(sigma u)/(exp(exp u)+1), calls it summable on the real line, and invokes Wiener’s theorem in the following convolution paragraph. The source’s selected bounded-solution clauses do not explicitly name a real or complex codomain, a measure, measurability, or almost-everywhere equality. Those modern conventions must be made explicit when stating the formal theorem; they are not quoted as printed words of Salem.

The positive-half-line formulation

The original A069 atlas clause, at https://github.com/the-omega-institute/trureturing/blob/915a86bf19ec91fdbd690a70e75c84014d237b7d/docs/develop/theory/RH_RESEARCH_LANE_THEORY.md#L8094, uses every delta in (1/2,1), every bounded measurable complex function on the positive half-line, and the equation

[ I_{\delta,f}(x)=\int_0^\infty \frac{t^{\delta-1}f(t)}{e^{xt}+1},dt=0\qquad(x>0). ]

Here powers of positive real t are real powers, then included in the complex numbers. Uniqueness means f = 0 almost everywhere for Lebesgue measure. The conjunction over all these delta is equivalent to the standard Riemann hypothesis. Values of f outside the positive half-line are immaterial.

Set mu = volume restricted to (0,infinity). The formal hypothesis NullMeasurable f mu is exactly measurability on the completed sigma algebra NullMeasurableSpace. Its completed measure has the same value on every set and exactly the same almost-everywhere filter as mu. Since the complex numbers are separable, this hypothesis is equivalent to AEStronglyMeasurable f mu. Trimming the completed measure back to the Borel sigma algebra gives mu; the Bochner integral trim theorem then identifies the two integrals for each such integrand. Thus arbitrary completed-measurable representatives are allowed without a Borel measurability hypothesis.

The maps v -> exp(-v) and t -> -log(t) preserve null sets in the respective directions. This follows from differentiability of the inverse maps on their domains and the theorem that a differentiable map sends a Lebesgue-null set to a null set. Positivity supplies the exp/log inverse identities. It follows that g(v) = f(exp(-v)) is almost-everywhere strongly measurable and that AE vanishing of g is equivalent to AE vanishing of f on (0,infinity).

Declaration correspondence and proof

The definition salemKernel is Salem’s page-1128 kernel included in the complex numbers. The theorem salem_integral_eq_convolution gives

[ I_{\delta,f}(e^y)=e^{-\delta y} \int_{\mathbb R}K_\delta(u)f(e^{u-y}),du. ]

This change-of-variables identity holds even for totalized integrals and is not, by itself, a convergence assertion. Its proof applies the existing MellinDilationFlow positive-logarithmic identity and translation invariance. For delta > 0 and bounded completed-measurable f, both sides are genuinely integrable: FermiMellin supplies positive-scale Mellin integrability; bounded multiplication controls the original integrand; the exponential Jacobian integrability equivalence and bounded multiplication control the convolution. A private convergent transport also identifies the original integral on the completed measure.

For salem_bounded_measurable_uniqueness_iff_rh, the forward implication uses the Fourier convention exp(-2 pi i u xi), so

[ \widehat K_\delta(\xi)=M(\delta,-2\pi\xi),\qquad M(\delta,\gamma)=\int_0^\infty \frac{t^{\delta-1+i\gamma}}{e^t+1},dt. ]

The existing FermiMellin criterion makes this transform nowhere zero under RH. Its product formula Gamma(s)(1-2^(1-s))zeta(s) is holomorphic in 0 < Re(s) < 1. Restricting scalars to the reals and composing with s = delta - 2 pi i xi gives the smoothness needed by the existing SmoothConvolutionUniqueness theorem. Applying it to g and transporting the AE conclusion proves original-domain uniqueness.

Conversely, a Mellin zero at delta + i gamma gives the actual function f_gamma(t) = exp(i gamma log t). It is measurable and has norm one; the interval (1,2] has positive measure, so this function cannot vanish almost everywhere. Its integrals are genuinely integrable. Positive Mellin scaling multiplies the zero at scale one by x^(-delta-i gamma), hence its original integral is zero for every x > 0. Uniqueness rules out every such zero, and the existing FermiMellin equivalence gives RH.

Implementation sources

These are applications of existing repository declarations, not transplants of the original article’s proof. FermiMellin’s convergence/product proof and its source adaptations are attributed separately in the Fermi Mellin note. The generic convolution cancellation theorem and its classical Wiener-density interpretation are documented in the Wiener note. That note uses Fulsche–Luef–Werner, arXiv:2405.08678v2, Theorem 2.1, printed page 3, with the group/Fourier/duality conventions on pages 2–4. The source’s angular frequency is 2 pi xi; its density theorem does not require the extra smoothness used by the repository’s cancellation proof. The original Wiener 1932 body and the published JFA version were not read. Salem attribution, this modern statement interpretation, and these Lean implementation sources therefore have distinct scopes.