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Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis

Assuming the simplicity of the Riemann zeta function's zeros, this paper derives an explicit formula for the discrete Laplace transform of the Möbius function involving double poles from colliding Gamma and zeta zeros, establishes a criterion for the Riemann hypothesis based on the transform's decay rate, and introduces special entire functions with closed-form Möbius-weighted Bessel series.

Original authors: Sergey Liflandsky

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sergey Liflandsky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Riemann zeta function, $\zeta(s)$ , as a giant, invisible musical instrument that holds the secret code to how prime numbers are scattered across the universe. For over 150 years, mathematicians have been trying to hear the exact melody of this instrument, specifically to prove the Riemann Hypothesis: the idea that all the "hidden notes" (the non-trivial zeros) of this instrument are perfectly tuned to a single, specific frequency line.

In this paper, Sergey Liflandsky picks up a new tool to listen to this music: the Discrete Laplace Transform of the Möbius function. Think of the Möbius function, $\mu(n)$ , as a cosmic switchboard operator that flips signs (+1, -1, or 0) based on the prime factors of a number. Liflandsky is asking: "What happens if we take all these switches, multiply them by a decaying signal ( $e^{-nx}$ ), and add them all up?"

The Big Discovery: A Collision of Worlds

The paper's main finding is a brand-new, explicit formula for this sum. It's like finding a master key that unlocks the sum using two different sets of information: the values of the zeta function at specific points and the locations of its hidden zeros.

Here is the twist that makes this paper special. In previous formulas (like the one for the Mertens function), the math was relatively clean. The "poles" (places where the math blows up) of one part of the equation and the "zeros" (places where the equation vanishes) of another part stayed apart.

But in this new formula, they crash into each other.

Liflandsky shows that the poles of the Gamma function (a famous mathematical helper) collide with the "trivial zeros" of the zeta function. It's like two trains on a collision course that don't just stop; they merge into a double pole. This collision creates a brand-new, messy, and beautiful term in the formula: a logarithmic term involving $\log(2\pi x)$ . This structure has never been seen before in these types of formulas. It's a unique fingerprint left by the collision of these mathematical forces.

The "Bessel" Surprise

The paper also introduces some "special entire functions" (mathematical objects that are smooth everywhere) named $\kappa$ , $\beta$ , $\lambda$ , and $\upsilon$ . These aren't just random names; they are the mathematical residue of that collision.

Liflandsky proves something surprising about these functions: they can be written as absolutely convergent series of Bessel functions. To visualize this, imagine the Bessel function as a ripple in a pond. The paper shows that these special functions are actually a perfect, never-ending stack of these ripples, weighted by the Möbius function. It's like discovering that a complex, chaotic storm is actually made of perfectly ordered, rotating waves.

The Riemann Hypothesis Test

The most exciting part is how this formula acts as a criterion (a test) for the Riemann Hypothesis.

The One-Way Street (Proven): The paper proves that if the sum of these Möbius switches behaves in a very specific way (shrinking at a rate of $O(x^{-1/2})$ as $x$ gets tiny), then the Riemann Hypothesis is definitely true. This direction requires no extra assumptions; it's a hard mathematical fact. If you see the sum shrinking that fast, the zeros must be on the critical line.
The Two-Way Street (Conditional): The paper argues that the reverse is also true: if the Riemann Hypothesis is true (and we assume the zeros are simple and don't get too close to zero), then the sum will shrink at that specific rate. However, this direction relies on a few extra assumptions about how the zeros behave, which are widely believed but not yet proven.

What the Paper Rules Out

The paper explicitly rules out the idea that this new formula is just a minor tweak of the old ones. It emphasizes that the logarithmic term ( $\log(2\pi x)$ ) is a structural feature that cannot be found in the classical formulas for the Mertens function. The old formulas had simple poles; this one has double poles. The math is fundamentally different.

How Sure Are We?

The Formula: The explicit formula (1.18) is proven, but it relies on the assumption that all non-trivial zeros are "simple" (meaning no two zeros sit exactly on top of each other). This is a standard assumption supported by massive numerical calculations (up to $3 \cdot 10^{12}$ ), but it hasn't been mathematically proven yet. If a zero were "double," the formula would need a slight adjustment, but the core logic would still hold.
The Criterion: The direction "Sum shrinks fast $\rightarrow$ Riemann Hypothesis is true" is unconditionally proven. No extra guesses are needed.
The Reverse: The direction "Riemann Hypothesis is true $\rightarrow$ Sum shrinks fast" is conditional. It holds if we assume the zeros are simple and don't get too close to zero (Hypothesis H).

The Bottom Line

Liflandsky hasn't solved the Riemann Hypothesis yet. Instead, he has built a new, highly sensitive detector. He found that the way the Möbius function decays is deeply tied to the geometry of the zeta function's zeros. The discovery of the double pole and the resulting logarithmic term is a new piece of the puzzle that was previously invisible. It's a rigorous, proven step forward that offers a fresh, unconditional way to test if the Riemann Hypothesis is true, provided we can measure that sum accurately enough.

Technical Summary: Explicit Formula for the Discrete Laplace Transform of the Möbius Function

Problem Statement
The paper addresses the derivation of an explicit formula for the discrete Laplace transform of the Möbius function, defined as $\Phi(e^{-t}) = \sum_{n=1}^\infty \mu(n)e^{-nt}$ . While explicit formulas for the Mertens function (the partial sums of $\mu(n)$ ) are well-established (e.g., Titchmarsh's Theorem 14.27), the behavior of the Laplace transform under the assumption of simple zeros of the Riemann zeta function presents unique analytical challenges. Specifically, the author investigates how the poles of the Gamma function $\Gamma(s)$ interact with the trivial zeros of the Riemann zeta function $\zeta(s)$ in the context of the Mellin inversion of $\Gamma(s)/\zeta(s)$ .

Methodology
The core methodology relies on the Mellin transform and contour integration techniques, following the classical approach of Hardy and Littlewood.

Mellin Representation: The author establishes the identity $\Gamma(s)/\zeta(s) = \int_0^\infty \Phi(e^{-t})t^{s-1}dt$ for $\text{Re}(s) > 1$ . By the Mellin inversion theorem, $\Phi(e^{-x})$ is represented as a contour integral of $\Gamma(s)x^{-s}/\zeta(s)$ .
Contour Shift: The integration contour is shifted from the line $\text{Re}(s) = \sigma > 1$ $Re (s) = σ > 1$ to the left, towards $-\infty$ $- \infty$ . This process involves calculating the residues of the integrand at its singularities:
- Non-trivial zeros ( $\rho$ ): Assuming Hypothesis (S) (all non-trivial zeros are simple), these contribute simple poles.
- Trivial zeros: The paper identifies a structural novelty here. The trivial zeros of $\zeta(s)$ occur at negative even integers ( $s = -2, -4, \dots$ ), which coincide with the poles of $\Gamma(s)$ . This collision creates double poles in the integrand, rather than the simple poles found in standard Mertens formulas.
- Odd negative integers: These correspond to simple poles where $\Gamma(s)$ has a pole but $\zeta(s)$ is regular and non-zero.
- Zero at $s=0$ : A simple pole contribution.
Residue Calculation:
- The residues at the double poles (negative even integers) are computed using Laurent expansions and logarithmic derivatives, revealing terms involving $\log(2\pi x)$ , the digamma function $\psi$ , and derivatives of the zeta function.
- The residues at the non-trivial zeros are calculated using L'Hôpital's rule under the assumption of simplicity.
Convergence and Bounds: To justify the contour shift, the author employs a uniform lower bound for $|\Gamma(s)|$ on moving vertical lines (Lemma 5.1), which is crucial for controlling the integral on the left edge of the contour where standard Stirling asymptotics fail. Lemma 5.2 (Titchmarsh) is used to select horizontal lines avoiding the zeros of $\zeta(s)$ .
Series Transformation: The contributions from the trivial zeros (the "entire functions" part of the formula) are re-expressed as absolutely convergent series of Bessel functions with rotated arguments, utilizing the power series definitions of $\kappa, \beta, \lambda, \upsilon$ .

Key Contributions and Results

Explicit Formula (Theorem 1.1):
Under the assumption that all non-trivial zeros of $\zeta(s)$ are simple (Hypothesis S), the paper derives the following explicit formula for $x > 0$ :
$\sum_{n=1}^\infty \mu(n)e^{-nx} = \sum_{\rho} \frac{\Gamma(\rho)x^{-\rho}}{\zeta'(\rho)} + \pi\kappa(x) + 2\lambda(x) + 4\upsilon(x) - 2\beta(x)\log(2\pi x) - 2$
where the sum over $\rho$ is a symmetric limit over non-trivial zeros. The functions $\kappa, \beta, \lambda, \upsilon$ are entire functions defined by power series involving $\zeta(2m+1)$ , $\zeta(2m+2)$ , and their derivatives.
- Structural Novelty: The term $-2\beta(x)\log(2\pi x)$ arises specifically from the double poles at the negative even integers. This logarithmic structure is absent in the classical explicit formula for the Mertens function, where trivial zeros contribute only simple poles.
Möbius–Bessel Expansions (Theorem 1.2):
The paper provides closed-form expressions for the entire functions $\kappa, \beta, \lambda, \upsilon$ as absolutely convergent Möbius-weighted series of Bessel functions $J_0$ with rotated arguments ( $2e^{-i\pi/4}\sqrt{2\pi x/k}$ ). For example:
$\beta(x) + i\kappa(x) = \sum_{k=1}^\infty \frac{\mu(k)}{k} \left( J_0\left(2e^{-i\pi/4}\sqrt{\frac{2\pi x}{k}}\right) - 1 \right)$
This establishes a direct link between the trivial zero contributions and Bessel-type functions analogous to those in Riesz and Hardy-Littlewood criteria.
Criterion for the Riemann Hypothesis (Theorem 1.3):
The paper establishes a criterion linking the asymptotic behavior of $\Phi(e^{-x})$ as $x \to 0^+$ to the Riemann Hypothesis (RH):
- Unconditional Direction: If $\sum \mu(n)e^{-nx} = O(x^{-1/2})$ as $x \to 0^+$ , then the Riemann Hypothesis is true. This direction requires no assumptions about the simplicity of zeros or the convergence of the zero sum.
- Converse Direction: If RH holds, along with Hypothesis (S) (simple zeros) and Hypothesis (H) (absolute convergence of the sum $\sum |\Gamma(\rho)/\zeta'(\rho)|$ ), then the bound $O(x^{-1/2})$ holds.

Significance and Claims
The paper claims three primary contributions:

The Exact Explicit Formula: It presents a formula where the collision of $\Gamma(s)$ poles with trivial zeros generates double poles and a distinct logarithmic term ( $-2\beta(x)\log(2\pi x)$ ), a feature not present in previous Mertens-type formulas.
Closed Forms for Special Functions: It demonstrates that the functions arising from trivial zeros admit absolutely convergent representations as Möbius-weighted Bessel series, extending the analogy with classical criteria.
A Refined Criterion: It provides a criterion for the Riemann Hypothesis where the "if" direction is unconditional. The proof relies on a new uniform lower bound for the Gamma function on moving lines (Lemma 5.1), which replaces fixed-abscissa asymptotics that fail in this specific contour deformation context.

The author notes that the method is robust: if Hypothesis (S) is false, the formula holds unconditionally with modified residue terms accounting for zero multiplicities. The paper also suggests that these techniques extend to Dirichlet L-functions and that the logarithmic terms appear generally for rational exponents in the kernel $e^{-n^\alpha x}$ .