Will a physicist prove the Riemann Hypothesis?

2014/10/31 by Marek Wolf · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #Mathematical physics #Mathematics #Partition function (quantum field theory) #Physics #Pure mathematics #Quantum mechanics #Random matrix #Random walk #Riemann hypothesis #Riemann surface #Riemann zeta function #Statistical physics #Theoretical physics #math-ph #math.MP #math.NT

paper · pdf · doi:10.1088/1361-6633/ab3de7

published as Reports on Progress in Physics, (2020) v.83, n.3, p. 036001 · Many changes in this version. Hamiltonian for trivial zeros of zeta is proposed

arxiv created 2015/12/01 · openalex created_date 2016/06/24 · openalex publication_date 2019/08/22 · arxiv updated 2020/02/25 · openalex updated_date 2026/08/05

Abstract

In the first part we present the number theoretical properties of the Riemann zeta function and formulate the Riemann Hypothesis. In the second part we review some physical problems related to this hypothesis: the links with Random Matrix Theory, relation with the Lee--Yang theorem on the zeros of the partition function, random walks, billiards etc.

Citations

• On the difference π(x) - li(x)
• A sharp region where 𝜋(𝑥)-fc(r) is positive
• The ternary Goldbach conjecture is true
• On the Hamiltonian whose spectrum coincides with the set of primes
• The Quantum Mechanical Potential for the Prime Numbers
• A new bound for the smallest 𝑥 with 𝜋(𝑥)>fc(r)
• On some reasons for doubting the Riemann hypothesis
• The quantum Mellin transform
• Sur la distribution des nombres premiers
• Open Circular Billiards and the Riemann Hypothesis
• Some Calculations of the Riemann Zeta-Function
• Riemannian quantum circuit
• Quantum mechanical potentials related to the prime numbers and Riemann zeros
• Computing π (x) analytically
• The Spectrum of Riemannium
• The Positivity of a Sequence of Numbers and the Riemann Hypothesis
• The First 50 Million Prime Numbers
• Nuclei, Primes and the Random Matrix Connection
• Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes
• Abundant Numbers and the Riemann Hypothesis
• The prime-counting function and its analytic approximations
• An Elementary Problem Equivalent to the Riemann Hypothesis
• Note sur les zéros de la fonction ζ(s) de Riemann
• On the Difference π( x )-li( x ) (I)
• On the Difference π( x ) − lix (II)
• Mathematical problems
• Prime Numbers and Computer Methods for Factorization

Cited by

• Hilbert–Pólya conjecture [wikipedia]

Related

• Physics of the Riemann Hypothesis
• Three Lectures on the Riemann Zeta-Function
• Currently there are no reasons to doubt the Riemann Hypothesis: The zeta function beyond the realm of computation
• The Riemann Hypothesis
• The Riemann Hypothesis and the possible proof