Riesz Function: Analysis & Applications

• The Riesz function is a special function and operator family linked to Dirichlet series, Möbius weights, and integral transforms in analytic number theory and harmonic analysis.
• Its series and transform representations, including Möbius-weighted and Mellin integral forms, offer rapid convergence and connections to the Riemann hypothesis.
• Key applications encompass spectral multipliers and singular integral operators, bridging analytic number theory with advanced harmonic and operator theory.

The Riesz function, originally introduced by Marcel Riesz and subsequently extended in a number of analytic and harmonic analysis contexts, designates a set of special functions and operators closely tied to the structure of Dirichlet series, weighted summability, Hardy spaces, operator theory, and analytic number theory. Its pivotal role spans spectral multipliers, harmonic analysis associated to special function expansions, and connections to the Riemann hypothesis via explicit series or integral representations. The precise form and usage of a “Riesz function” depend on the structural setting, but typical incarnations involve entire series tied to Möbius weights, transformations via orthogonal polynomial systems, or operator-theoretic constructions defining singular integral transforms in non-classical frameworks.

1. Classical Riesz Function: Definition and Series Representations

The prototypical Riesz function is given as an entire function defined by a Maclaurin series whose coefficients encode significant arithmetic information. For the classical case considered by Marcel Riesz, one has

$$R(x) = \sum_{k=1}^\infty \frac{(-1)^{k+1} x^k}{(k-1)! \cdot \alpha(2k)}$$

where the denominator involves expression in terms of Bernoulli numbers: $$\alpha(2k) = (-1)^{k+1} B_{2k} \frac{2^{2k}}{(2k)!}$$ yielding, after simplification, a rapidly growing/inverson coefficient structure. This series (and its generalizations):

• converges for all $x \in \mathbb{C}$
• can be recast via acceleration schemes, notably converting Dirichlet series $\sum_n a_n n^{-s}$ into rapidly convergent entire function expansions by

$$R_c(x) = \sum_{k=1}^\infty \frac{(-1)^k f(ck) x^k}{(k-1)!}$$

with $f(s)$ the sum of the Dirichlet series in some right half-plane.

A Möbius-weighted variant crucially connected to the distribution of primes and the Riemann zeta function is

$$R(x) = \sum_{n=1}^\infty \frac{\mu(n)}{n^2} e^{-x/n^2}$$

where $\mu(n)$ is the Möbius function. This analytic object admits inverse Mellin transforms, and via appropriate integral transforms (e.g., using the kernel $y^{-1/2} e^{-x^2/4y}$), connects to Laplace transforms of Dirichlet-type series involving Möbius weights (Patkowski, 19 Aug 2025). This analytic flexibility enables the Riesz function to encode deep arithmetic information in analytic form.

2. Riesz Function and the Riemann Hypothesis

The Riesz function is deeply intertwined with the Riemann hypothesis (RH). Explicitly, its analytic growth encapsulates zero-free regions of the zeta function. The classical result, via Mellin transform, shows: $$\int_0^\infty R_c(x) x^{s-1} dx = \frac{f(-cs)\Gamma(s)}{\zeta(-2s)}$$ where $R_c(x)$ is the Riesz function constructed from a Dirichlet series with suitable parameter $c > 1$, and $\zeta(s)$ is the Riemann zeta function. The location of the singularities of this Mellin transform is thus compatible with the regions free of nontrivial zeros of $\zeta(s)$.

Marcel Riesz’s key insight, proved and extended in later works (Smith, 2012), is that the RH is equivalent to

$R_c(x) = O(x^{2c+\epsilon})$

as $x \to \infty$ for every $c > 1$ and all $\epsilon > 0$. Informally, the Riemann hypothesis holds if and only if the Riesz function exhibits sufficiently slow growth on the real axis.

Generalizations include asymptotic expansions involving the nontrivial zeros $\rho$ of $\zeta(s)$; in the Riesz function’s large–$x$ asymptotics, the leading oscillatory terms are

$$R(x) \sim \sum_{\rho} \frac{\pi \csc(\pi\rho/2) 2 K'(\rho) x^{\rho/2}}{\zeta'(\rho)}$$

where $K(s)$ is an auxiliary function constructed from the functional equation, and the sum is over nontrivial zeros.

3. Integral and Transform Representations

A fundamental property of the Riesz function is the existence of explicit integral representations that facilitate analytic continuation and paper of zeros. For example,

$$R(x) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{x^{-s} \Gamma(s)}{\zeta(2 - 2s)} ds$$

for suitable vertical lines $\text{Re}(s) = c$.

A further key formula, as established in (Patkowski, 19 Aug 2025), provides a bridge between series involving $\mu(n)/n$ and exponentially weighted versions of the Riesz function: $$\frac{\pi}{2} \sum_{n=1}^\infty \frac{\mu(n)}{n} e^{-a x/n} = a \int_0^\infty \left(\sum_{n=1}^\infty \frac{\mu(n)}{n^2} e^{-a^2y/n^2}\right) \frac{\sqrt{\pi}}{2} y^{-1/2} e^{-x^2/4y} dy$$ This explicit transform quantifies how Laplace-type integrals of the prime-detection Möbius series are smoothened through the “heat” kernel into Riesz-type weighted exponential sums, thus connecting prime-counting sums, Möbius inversion, and analytic number theory directly to the Riesz function.

Fourier cosine integrals are prominent in related explicit formulas, facilitating the paper of zero distribution and oscillatory behavior. For instance,

$$R(x) = \int_0^\infty k(t) \cos\left(\frac{2\sqrt{t}}{Vx}\right) dt$$

with the concrete kernel given by Möbius-weighted exponentials.

4. Asymptotic Properties, Numerical Evaluation, and Decay Patterns

Numerical investigations (Paris, 2021) confirm the theoretically predicted asymptotics of generalized Riesz functions

$$S_{m,p}(x) = \sum_{k=0}^\infty \frac{(-1)^{k-1} x^k}{k! \zeta(mk+p)}$$

for large $x$. In the classical cases (e.g., $m=2$, $p=1$, corresponding to Hardy–Littlewood’s version, and $m=p=2$, the original Riesz function), the decay pattern is: $$S_{2,1}(x) = O(x^{-1/4+\epsilon}), \qquad S_{2,2}(x) = O(x^{-3/4+\epsilon})$$ for $x$ large. These rates are necessary and sufficient conditions for the truth of the Riemann hypothesis. The functions display superimposed oscillations reflecting contributions from sums over zeros $\rho$ of $\zeta(s)$, i.e.,

$$S_{2,2}(x) \sim x^{-3/4} \sum_{k} |A_k| \cos\left( \frac{\gamma_k}{2} \log(x) + \psi_k\right )$$

where $\gamma_k$ is the imaginary part of nontrivial zeros.

Efficient numerical computation schemes exploit Möbius expansion, decomposition of terms with exponentially small cutoffs, and acceleration by Kummer’s transformation and hypergeometric recursions, achieving rapid convergence for large arguments.

5. Connections to Summability, Spectral Theory, and Harmonic Analysis

The “Riesz function” terminology encompasses a broader class of operators and kernels, especially in the context of harmonic analysis, spectral multipliers, and summability methods.

• In the context of Laguerre expansions, the Riesz transform is defined spectrally as

$Rf = \sqrt{\pi} \frac{d}{dx} L^{-1/2}f$

where $L$ is a differential operator encoding the Laguerre structure. The mapping properties of this operator serve to characterize atomic Hardy spaces associated with Laguerre expansions: $f \in H^1_{\text{at}}(X)$ if and only if $f$ and $Rf$ are integrable (Preisner, 2010).

• In spectral theory for Schrödinger operators, the Riesz transforms $R_j = \frac{\partial}{\partial x_j} L^{-1/2}$ (with $L$ a generalized Schrödinger operator) provide a full characterization of associated Hardy spaces, underlining their universal role as probes of regularity and scale (Dziubański et al., 2010).
• Through integral operator and kernel representations, the Riesz function appears in generalized summation and smoothing procedures, such as Riesz means for series and spectral expansions, with kernel smoothing parameters adapting to the geometry or operator under paper.

6. Number-Theoretic and Arithmetic Connections

The Riesz function interlocks with other core arithmetic objects:

• The Riesz criteria for the RH is equivalent to bounds on partial sums involving the Möbius function, encapsulating connections to the Mertens function and zero-detection via oscillatory exponential sums (Patkowski, 2015).
• Explicit formulas involving integration or series manipulations expose direct relationships between Möbius-weighted sums, Gram series for the prime-counting function, and the analytic structure of the Riesz function (Patkowski, 19 Aug 2025).
• Mellin and Fourier-analytic representations integrate the Riesz function into the web of analytic number theory, offering tools for exploring explicit formulas, oscillatory phenomena, and error terms in prime-counting and related arithmetic summations.

7. Broader Contexts: Riesz Operators, Potentials, and Frames

In advanced harmonic analysis, the “Riesz function” label admits substantial operator-theoretic generalization:

• In the Dunkl transform setting, Riesz potentials act as fractional integral operators, exhibiting weighted boundedness (Stein–Weiss inequalities) and sharp constants in generalized function space frameworks (Gorbachev et al., 2017).
• In time-frequency analysis and the theory of frames, Riesz sequences and related counting function asymptotics structure density criteria and local distribution for coherent systems in locally compact groups, providing necessary density bounds and outlining the relationship of Riesz sequences to the volume growth and localization of generating vectors (Papageorgiou et al., 18 Jul 2024).

This diversity illustrates the centrality of Riesz-type functions and operators in both analysis and number theory, with applications ranging from characterizing singular integral operators and function spaces, to encoding arithmetical conjectures and providing analytic machinery for tackling central open problems.

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In summary, the Riesz function encapsulates a rich interplay between analytic number theory, spectral theory, and harmonic analysis, offering both deep structural insights and practical tools for analyzing zeros, summability, and integral transforms tied to fundamental mathematical problems. Its analytic growth, transform representations, and connections to the Möbius function provide critical criteria for central conjectures such as the Riemann hypothesis, while its operator-theoretic avatars inform the structure of function spaces and the boundedness of singular integral operators across several non-classical settings.