German Sierra
We present a brief review of the spectral approach to the Riemann hypothesis, according to which the imaginary part of the non trivial zeros of the zeta function are the eigenvalues of the Hamiltonian of a quantum mechanical system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is shown that the Riemann hypothesis holds for the Dirichlet beta function $L_{-4}(s)$ if and only if it holds for $ζ(s)$- a particular case of the Generalized Riemann Hypothesis.
Physics is a fertile environment for trying to solve some number theory problems. In particular, several tentative of linking the zeros of the Riemann-zeta function with physical phenomena were reported. In this work, the Riemann operator is introduced and used to transform the Riemann's hypothesis in a Heisenberg-type uncertainty relation, offering a new way for studying the zeros of Riemann's function
We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the …
We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $ζ(s)$.
We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{λ_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η_j are valid. The constants η_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, n…
Supersymmetric quantum mechanics and the Riemann hypothesis
本文构建超对称量子力学模型，使哈密顿量本征值表现为 Riemann zeta 函数乘积，并证明其平凡与非平凡零点对应消失的基态能量，从而为黎曼猜想提供自然的超对称表述。
The review of the mathematical treatment of plasmon resonances as an eigenvalue problem for specific boundary integral equations is presented and general properties of plasmon spectrum are outlined. Promising applications of plasmon resonances to magnetics are described. Interesting relation of eigenvalue treatment of plasmon resonances to the Riemann hypothesis is discussed.
The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n = 1,2,..., showing in detail that the leading behaviour (1/2) ln n of lambda_n/n is absent in c_n. Additional results are presented, including a novel explicit representation of c_n in terms of the Stieltjes constants gamma_j. We conjecture as to the large-n behaviour of c_n. Should this conjecture hold, validity of the Riemann hypothesis would follow.
Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Ri…
We propose an architecture of a conjecture concerning the Riemann Hypothesis in the form of an "alternative" to the Pólya strategy: we construct a Hamiltonian H_Polya whose spectrum coincides exactly with that of the Harmonic Oscillator Hamiltonian H_osc if and only if the Riemann Hypothesis holds true. In other words, it can be said that we formulate the Riemann Hypothesis by means of a non-commutative structure on the real axis, viz., that of the Harmonic Oscillator, by an equation of the type H_Polya(H_osc) = H_osc: the Harmonic Oscillator operator, if viewed as an argument of H_Polya, repr…
We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of t…
An analogue of the Riemann Hypothesis via quantum walks
该研究利用 Konno-Sato 定理，将图上量子游走与黎曼猜想类比联系起来，并通过完全图、圈图和星形图示例说明这一构造。
The aim of this note is to remove an implausible assumption in Moser's theorem \cite{JM} to establish our new theorem 1 which gives a lower estimate for the sum $p+c^2ρ$ on Riemann hypothesis. Corollary 1 gives a rather plausible construction of infinitely many models of universe with positive density($ρ$) and pressure($p$) since it makes use of the state equation in the form of an inequality.
We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta func…
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.
We present a Non-relativistic Quantum mechanical model, which exhibits the realization of Riemann Conjecture. The technique depends on exposing the $S$-wave Jost function at zero energy and in identifying it with the Riemann $ξ(s)$ function following a seminal paper of N. N. Khuri.
The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.
In recent years, there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis. Most of these kind of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory. In this letter we show that the real part of the non-trivial zeros of the Riemann zeta function extremizes the grand potential corresponding to a simple model of one-dimensional classical lattice gas, the critical point being located at 1/2 as the Riemann Hypothesi…
We invoke Carlson's theorem to justify and to confirm the results previously obtained on the validity of Riemann Hypothesis via the coupling constant spectrum of the zero energy S-wave Jost function a la N. N. Khuri, for the real, repulsive inverse-square potential in non-relativistic quantum mechanics in 3 dimensions.
We discuss new relations among string theory, four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) and the Riemann hypothesis. It is known that the Riemann hypothesis is equivalent to an inequality for the sum of divisors function $σ(n)$. Based on previous results in literature, we focus on the fact that $σ(n)$ appears in a problem of counting supersymmetric states in the $\mathcal{N}=4$ SYM with $SU(3)$ gauge group: the Schur limit of the superconformal index plays a role of a generating function of $σ(n)$. Then assuming the Riemann hypothesis gives bounds on information on…
This paper compares the distribution of zeros of the Riemann zeta function $ζ(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$. Criteria are described for constructing a suitable quotient function of these, with properties advantageous for establishing an accessible proof that $ζ(s)$ must also have all its zeros on the critical line: the celebrated Riemann hypothesis. While the argument put forward is not at the level of rigour required to constitute a full proof of the Riemann hypothesis…
The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the emergence of Poisson-like distributions and we present some numerical experiments which indicate that the RH may barely be true.
We present, using spectral analysis, a possible way to prove the Riemann's hypothesis (RH) that the only zeroes of the Riemann zeta-function are of the form s=1/2+iλ_n. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann's conjecture, inspired in the Hilbert-Polya proposal; it uses an inverse eigenvalue approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1-s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。