Beurling Zeta Functions

• Beurling zeta functions are analytic extensions of the Riemann zeta function using generalized primes and integers under regular asymptotic conditions.
• Recent work has established Carlson-type zero-density estimates and refined prime number theorem results, highlighting oscillatory error terms and optimal bounds.
• Analytic methods such as zero-detecting sums, contour integration, and probabilistic techniques extend classical approaches to these generalized number systems.

A Beurling zeta function is an analytic object associated to a generalized number system, extending the classical Riemann zeta function by replacing the integers and primes with more general sequences subject to regularity conditions. Developed originally by Arne Beurling, the theory provides a framework for investigating analogues of prime number theory in far more flexible algebraic and analytic settings. Recent research has focused on the analytic continuation, zero-density estimates, and oscillatory phenomena associated with Beurling zeta functions, revealing deep structural similarities and differences compared to the classical zeta and L-function theory.

1. Definition and Analytic Framework

Let $\mathcal{P}$ be a sequence of “generalized primes” and $\mathbb{N}_\mathcal{P}$ the associated multiplicative semigroup (“generalized integers”) formed by finite ordered products of elements of $\mathcal{P}$ with possible multiplicity. The corresponding counting function is $$N_\mathcal{P}(x) := \# \{ n \in \mathbb{N}_\mathcal{P} : n \leq x \}$$. A fundamental regularity assumption is a Beurling analogue of the classical integer asymptotic,

$N_\mathcal{P}(x) = A x + O(x^\theta)$

for some $A>0$, $0 \leq \theta < 1$.

The Beurling zeta function is defined by a Dirichlet series,

$$\zeta_\mathcal{P}(s) := \sum_{n \in \mathbb{N}_\mathcal{P}} n^{-s} = \prod_{p \in \mathcal{P}} (1 - p^{-s})^{-1} \qquad \Re(s) > 1,$$

or, equivalently, via the Mellin–Stieltjes transform,

$$\zeta_\mathcal{P}(s) = \int_1^\infty x^{-s} \, dN_\mathcal{P}(x).$$

The generalized von Mangoldt function $\Lambda_\mathcal{P}$ is defined by $\Lambda_\mathcal{P}(n) = \log p$ if $n = p^k$ for $p \in \mathcal{P}$ and $k \geq 1$, zero otherwise. The Chebyshev–Mangoldt summatory function is $$\psi_\mathcal{P}(x) = \sum_{n \leq x} \Lambda_\mathcal{P}(n)$$.

If $N_\mathcal{P}(x)$ satisfies the above regularity (commonly called Axiom A), then $\zeta_\mathcal{P}(s)$ has a meromorphic continuation to $\Re(s) > \theta$ with a simple pole at $s=1$ and controlled behavior in the critical strip $\theta < \Re(s) < 1$ (Révész, 2021).

2. Zero-Density Estimates

A central object of paper is the distribution of zeros of $\zeta_\mathcal{P}(s)$ in the critical region. In analogy with classical results (Titchmarsh, Selberg), the zero-count $N(\alpha, T)$ denotes the number of zeros $\rho = \beta + i\gamma$ of $\zeta_\mathcal{P}(s)$ with $\beta > \alpha$ and $|\gamma| < T$.

A key advance is the development of Carlson-type zero-density theorems for Beurling zeta functions. Under the main assumption

$N_\mathcal{P}(x) = A x + O(x^\theta),$

one obtains for every fixed $\alpha > \theta$ and large $T$,

$$N(\zeta_\mathcal{P}; \alpha, T) \ll T^{\frac{4(1-\alpha)}{3-2\alpha-\theta}}(\log T)^9$$

(Broucke, 16 Sep 2024). If additional moment or pointwise bounds for $\zeta_\mathcal{P}(1+it)$ are known, this can be improved to

$$N(\zeta_\mathcal{P}; \alpha, T) \ll T^{2k(1-\alpha) + o(1)}$$

for $L^{2k}$-bounds, or

$$N(\zeta_\mathcal{P}; \alpha, T) \ll_B T^{(1-\alpha)/(2(1+2B))} (\log T)^9$$

under subconvexity-type estimates for $|\zeta_\mathcal{P}(1+it)|$ (Broucke, 16 Sep 2024). This regime encompasses analogues of classical Titchmarsh and Ingham density theorems.

In cases where the “norm” on integers is integral and a pointwise Ramanujan-type condition holds (i.e., restricts the multiplicity $G(\nu)$ of representations as generalized integers), even stronger zero-density results approach the Density Hypothesis, with exponents tending to $2(1-\sigma)$ for $\sigma \to 1^-$ (Révész et al., 17 Jul 2024). The methodology often utilizes zero-detecting sums, kernel function techniques, and Halász' method rather than mean-value theorems for Dirichlet polynomials.

Estimate Type	Hypotheses	Exponent in $T$
General Carlson-type	Axiom A, $\theta \in [0,1)$	$\frac{4(1-\alpha)}{3-2\alpha-\theta}$
Moment/pointwise improvement	$L^{2k}$ or subconvexity bounds	$2k(1-\alpha)$ or $(1-\alpha)/(2(1+2B))$
Near Density Hypothesis	Axiom A, integrality, pointwise Ramanujan condition	$2(1-\sigma)$ (as $\sigma \to 1^-$)

These results are essential for applications to error terms in generalized prime-counting and have led to significant improvements over previous literature, pushing the constants in the exponents closer to their conjectural minima.

3. Prime Number Theorem and Oscillation of the Error Term

The Beurling prime number theorem (PNT) states that, under Axiom A,

$\psi_\mathcal{P}(x) \sim x \qquad (x \to \infty)$

is equivalent to the absence of zeros of $\zeta_\mathcal{P}(s)$ to the right of $\Re(s) = 1$ (Révész, 2021, Révész, 2022). Quantitative error terms are controlled via the explicit formula (Riemann–von Mangoldt type),

$$\psi_\mathcal{P}(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} + O(\cdots)$$

where the sum runs over (weighted) nontrivial zeros in the critical strip.

Oscillation phenomena for the remainder, $\Delta_\mathcal{P}(x) := \psi_\mathcal{P}(x) - x$, are directly linked to the presence of zeros. Specifically, for a nontrivial zero $\rho_0 = \beta_0 + i\gamma_0$,

$$\limsup_{x \to \infty} |\Delta_\mathcal{P}(x)| \approx (\pi/2)x^{\beta_0}/|\rho_0|$$

is attainable via careful analytic construction, including the use of low-norm sine polynomials and probabilistic prime selection methods (Révész, 2022). The constant $\pi/2$ emerges as an optimal bound due to interference among oscillatory zero-terms and is achieved in constructed generalized number systems using prime random approximation (Révész, 2022).

4. Structure of Generalized Systems and Methodologies

Much of the analytic apparatus developed for the classical Riemann zeta function translates, with adaptations, to the Beurling setting due to carefully controlled generalizations:

• Broken-line Contour Integration: For explicit prime-counting formulas, Perron's formula and factorization of the Beurling zeta function across specialized contours are central (Révész, 2021).
• Zero-detecting Sums and Kernel Methods: Modern proofs of zero-density results avoid intricate large-sieve inequalities and instead employ elementary yet powerful detection techniques, kernel smoothing, and functional transforms (Révész et al., 17 Jul 2024).
• Mean-Value Theorem Extensions: The classical mean-value theorem for Dirichlet polynomials is generalized to sequences of Beurling integers using “smoothing” and counting in short intervals (Broucke et al., 2022). This is crucial where the lack of regular additive structure in the generalized integers impedes direct application of traditional arguments.
• Beurling Algebra and Spectral Methods: The algebraic structure underlying Beurling zeta functions is tied to weighted group algebras and their spectra, facilitating interpretation of analytic properties via harmonic analysis (Basit et al., 2013).

5. Beurling Regular Variation and Asymptotic Representation

The analysis of large-scale behavior in counting functions and related arithmetic quantities often employs Beurling regular variation (Bingham et al., 2013). A function $f$ is $\varphi$-regularly varying if

$\frac{f(x + t\varphi(x))}{f(x)} \to e^{\rho t}$

for some drift index $\rho$, uniformly in $t$ as $x \to \infty$. This uniformity and the resulting representation

$$f(x) = d(x)\exp\left\{\rho \int_1^x \frac{du}{\varphi(u)}\right\}$$

with $d(x)$ slowly varying, provide key inputs for Tauberian theorems governing asymptotics and analytic continuation for Beurling zeta functions.

Such results extend the classical theory of regular and slow variation, enabling generalizations of the PNT and the paper of zero-free regions and growth estimates in the non-classical context (Bingham et al., 2013). The connection is explicit: Beurling regular variation often describes the dominant term in the counting function that determines the analytic properties and critical strip geometry of $\zeta_\mathcal{P}(s)$.

6. Probabilistic and Functional Criteria

Recent research has expanded the analytic framework to encompass probabilistic analogues of the Nyman–Beurling criterion for the Riemann hypothesis (Darses et al., 2018). In this approach, the classical dilation parameters used in function space approximation are replaced by random variables, typically with exponential or gamma distributions. Under mild hypotheses, these probabilistic criteria (pNB and gNB) are shown to be equivalent to the validity of the Riemann hypothesis, with the randomness providing new flexibilities and regularization techniques.

Explicitly, the general NB criterion with random dilations constrains the $L^2$ error between indicator functions and suitably averaged combinations of dilated fractional part functions, demonstrating that the presence of “typical” random structures suffices for the approximation property: $$\mathbb{E} \int_0^\infty \left[ \chi(t) - \sum_{k=1}^n c_{k,n}\left\{ \frac{Z_{k,n}}{t} \right\} \right]^2 dt \to 0$$ where $Z_{k,n}$ are parameterized random variables (Darses et al., 2018).

The probabilistic setting also facilitates precise control over tail behavior and regularizes the impact of non-compactly supported kernels, at a quantifiable cost in the size and decay of the approximation coefficients (the “price to pay”) (Darses et al., 2018).

7. Optimality, Open Questions, and Outlook

The landscape of zero-density estimates and error-term oscillations for Beurling zeta functions is now populated with results closely analogizing – and in special cases nearly matching – the best known classical bounds. The proved optimality of several zero-density exponents, conditioned on regularity and Ramanujan-type assumptions, leaves open the question of eliminating or weakening these constraints while retaining sharpness.

Ongoing research continues to refine the balance between elementary zero-detection methods (kernel functions, Halász' method), mean-value theorems, and probabilistic approximations, and to explore the consequences for the distribution of generalized primes in short intervals, the construction of systems with prescribed zeta zero loci, and the transfer of techniques between the Beurling and Selberg or L-function settings (Révész et al., 17 Jul 2024, Broucke et al., 2022, Broucke, 16 Sep 2024).

References to Key Results

Main Result	Paper (arXiv id)	Conditions
Explicit Riemann–von Mangoldt	(Révész, 2021)	Axiom A only
General Carlson zero-density	(Révész, 2022, Révész et al., 17 Jul 2024, Broucke, 16 Sep 2024)	Axiom A; further regularity as needed
Prime number theorem oscillation	(Révész, 2022)	Axiom A; explicit error amplitude
Beurling regular variation	(Bingham et al., 2013)	Baire/measurable; uniformity
Probabilistic NB criterion	(Darses et al., 2018)	Random dilation families

The general theory of Beurling zeta functions, tying together spectral algebra, regular variation, probabilistic analytic number theory, and explicit zero-density phenomena, now provides an incorporated analytic framework robust enough to transfer and sharpen many techniques and results from the classical prime number theory to abstract generalized systems.