A proof of Riemann's hypothesis via Hadamard-Weierstrass factorization

Abstract: Using the Hadamard-Weierstrass factorization theorem for Riemann's ξ function, we discuss and prove Riemann's hypothesis.

• The paper presents a direct proof of the Riemann Hypothesis by equating the standard Hadamard product with a version restricted to zeros on the critical line.
• It leverages Weierstrass factorization to build the xi(s) function and rigorously bounds the error between the full zero set and critical-line zeros.
• The approach implies all nontrivial zeros lie on the critical line, a result that impacts prime number theory and related fields.

A Proof of the Riemann Hypothesis via Hadamard-Weierstrass Factorization

Introduction and Problem Statement

The paper "A proof of Riemann's hypothesis via Hadamard-Weierstrass factorization" (2607.04338) proposes a direct proof of the Riemann Hypothesis (RH) by leveraging Hadamard's product representation for the Riemann $\xi$-function, which is an entire function intricately connected to the Riemann zeta function $\zeta(s)$. Instead of employing the machinery of $L$-functions or appealing to classical approaches in analytic number theory, the author isolates the analytic structure of $\xi(s)$, constructs an explicit infinite product over its nontrivial zeros, and compares this with a product over zeros hypothetically constrained to the critical line.

This work discusses both formal aspects---via factorization theorems due to Weierstrass and Hadamard---and technical convergence and bounding arguments ultimately designed to demonstrate the claimed equality $\xi(s)=\widehat{\xi}(s)$, where $\widehat{\xi}(s)$ is built assuming all zeros lie on $\operatorname{Re}(s)=\frac{1}{2}$.

Analytic Structure of $\xi(s)$ and Critical Strip Zeros

The Riemann zeta function $\zeta(s)$ admits a meromorphic extension to $\mathbb{C}$ with a sole simple pole at $\zeta(s)$0 and satisfies the functional equation:

$\zeta(s)$1

for $\zeta(s)$2. The paper notes the trivial zeros at $\zeta(s)$3 and focuses on the nontrivial zeros within the open strip $\zeta(s)$4. The distribution and simplicity of zeros are discussed in view of known results (i.e., infinitely many zeros on the critical line, all zeros so far discovered appear simple).

The entire function $\zeta(s)$5 is constructed as

$\zeta(s)$6

which satisfies $\zeta(s)$7, and its zeros coincide with the nontrivial zeros of $\zeta(s)$8.

Hadamard’s Product and Weierstrass Factorization

The classical Weierstrass Factorization Theorem guarantees a convergent factorization of any entire function in terms of its zeros, possibly multiplied by an exponential. For $\zeta(s)$9, Hadamard supplied the canonical factorization:

$L$0

where the product runs over all nontrivial zeros $L$1. A more symmetric formulation explicitly pairs zeros with $L$2 as:

$L$3

where $L$4 runs over zeros in the upper half of the critical strip. For zeros lying on the critical line, i.e., $L$5, this further reduces to

$L$6

The paper highlights that the absolute convergence of the product is guaranteed by standard estimates on the zeros (especially their imaginary parts tending to infinity and a summability condition on their inverses squared), and the functional equation is preserved.

Main Theorem and Reduction to Zeros on the Critical Line

Central claim: If one can exhibit a sequence $L$7, with $L$8 strictly increasing and $L$9, such that the zeros of $\xi(s)$0 coincide with $\xi(s)$1, then the Hadamard product for $\xi(s)$2 coincides with the "critical line" product

$\xi(s)$3

for all $\xi(s)$4. The paper thus reduces RH to verifying $\xi(s)$5, where $\xi(s)$6 is built under the RH assumption.

A sequence of conditions (C1)–(C5) is introduced to formalize the allowable discrepancy between "true" zeros and "critical line" zeros, with the error width $\xi(s)$7 as $\xi(s)$8; convergence conditions are established to ensure the involved products are entire and match $\xi(s)$9.

Analytical Bounds and Error Estimates

The technical core is a series of lemmas bounding the difference between the partial products over true zeros (arbitrary in the critical strip) and those restricted to the critical line. Employing a telescoping product-difference inequality and explicit algebraic manipulation of the quadratic terms $\xi(s)=\widehat{\xi}(s)$0 and $\xi(s)=\widehat{\xi}(s)$1, error bounds are given by:

$\xi(s)=\widehat{\xi}(s)$2

with detailed explicit expressions for these terms under the given zero structure.

Uniform convergence and total error control are demonstrated in the limit $\xi(s)=\widehat{\xi}(s)$3, under the hypotheses, with precise manipulations of the bounding sequences and their relations to the locations of the zeros.

Convergence and Uniformity

By invoking completeness arguments for compact subsets of the critical strip and exploiting the summability of the error bounds, the analysis demonstrates that the infinite product for $\xi(s)=\widehat{\xi}(s)$4 converges absolutely and uniformly to the entire function $\xi(s)=\widehat{\xi}(s)$5 for all $\xi(s)=\widehat{\xi}(s)$6. Thus, the only possible zeros are at $\xi(s)=\widehat{\xi}(s)$7, establishing that all nontrivial zeros of $\xi(s)=\widehat{\xi}(s)$8 are on the critical line.

Implications and Perspectives

If valid, the result addresses the central open problem in analytic number theory, namely RH, by an entirely analytic route, circumventing traditional approaches relying on the deep theory of $\xi(s)=\widehat{\xi}(s)$9-functions, random matrix analogies, or advanced spectral methods. The proof structure highlights the power of infinite product representations, careful bounding, and functional equations in entire function theory.

The theoretical implication is that the locations of zeros for $\widehat{\xi}(s)$0 are rigidly determined by the analytic structure of $\widehat{\xi}(s)$1 once the constraints for entire functions with the prescribed growth are in place. Practically, affirming RH would corroborate many contingent results in prime number theory, error terms in the prime number theorem, and various results in random matrix theory and mathematical physics.

It is worth noting, however, that such attempts at elementary (analytic) proofs must withstand extreme scrutiny, particularly in controlling uniformity and limiting procedures at each technical step.

Conclusion

The paper develops a detailed analytic argument, constructing the Riemann $\widehat{\xi}(s)$2-function via the Hadamard-Weierstrass factorization and rigorously bounding the error between products over general and critical line zeros. The main claim---all nontrivial zeros of $\widehat{\xi}(s)$3 lie on $\widehat{\xi}(s)$4---is deduced by comparing these representations under strict summability and approximation conditions. The approach is formal, employing established complex analysis and entire function theory, and the implications for number theory and related fields are substantial, conditioned on the correctness of the technical bounding and convergence arguments.