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Register a new userMany-Body Fermions and Riemann Hypothesis
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We study the algebraic structure of the eigenvalues of a Hamiltonian that corresponds to a many-body fermionic system. As the Hamiltonian is quadratic in fermion creation and/or annihilation operators, the system is exactly integrable and the complete single fermion excitation energy spectrum is constructed using the non-interacting fermions that are eigenstates of the quadratic matrix related to the system Hamiltonian. Connection to the Riemann Hypothesis is discussed.
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In this article, we will prove Riemann Hypothesis by using the mean value theorem of integrals. The function $ xi(s) $ is introduced by Riemann, which zeros are identical equal to non-trivial zeros of zeta function.The function $ xi(s) $ is an entire function, and its real part and imaginary part can be represented as infinite integral form. In the special condition, the mean value theorem of integrals is established for infinite integral. Using the mean value theorem of integrals and the isolation of zeros of analytic function, we determined that all zeros of the function $ xi(s) $ have real part equal to$frac{1}{2}$, namely, all non-trivial zeros of zeta function lies on the critical line. Riemann Hypothesis is true.
Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1 - s)|. We have shown that sigma = 1/2 is the only numeric solution that satisfies this requirement.
The Riemann hypothesis is equivalent to the $varpi$-form of the prime number theorem as $varpi(x) =O(xsp{1/2} logsp{2} x)$, where $varpi(x) =sumsb{nle x} bigl(Lambda(n) -1big)$ with the sum running through the set of all natural integers. Let ${mathsf Z}(s) = -tfrac{zetasp{prime}(s)}{zeta(s)} -zeta(s)$. We use the classical integral formula for the Heaviside function in the form of ${mathsf H}(x) =intsb{m -iinfty} sp{m +iinfty} tfrac{xsp{s}}{s} dd s$ where $m >0$, and ${mathsf H}(x)$ is 0 when $tfrac{1}{2} <x <1$, $tfrac{1}{2}$ when $x=1$, and 1 when $x >1$. However, we diverge from the literature by applying Cauchys residue theorem to the function ${mathsf Z}(s) cdot tfrac{xsp{s}} {s}$, rather than $-tfrac{zetasp{prime}(s)} {zeta(s)} cdot tfrac{xsp{s}}{s}$, so that we may utilize the formula for $tfrac{1}{2}< m <1$, under certain conditions. Starting with the estimate on $varpi(x)$ from the trivial zero-free region $sigma >1$ of ${mathsf Z}(s)$, we use induction to reduce the size of the exponent $theta$ in $varpi(x) =O(xsp{theta} logsp{2} x)$, while we also use induction on $x$ when $theta$ is fixed. We prove that the Riemann hypothesis is valid under the assumptions of the explicit strong density hypothesis and the Lindelof hypothesis recently proven, via a result of the implication on the zero free regions from the remainder terms of the prime number theorem by the power sum method of Turan.
One of the key tasks in physics is to perform measurements in order to determine the state of a system. Often, measurements are aimed at determining the values of physical parameters, but one can also ask simpler questions, such as is the system in state A or state B?. In quantum mechanics, the latter type of measurements can be studied and optimized using the framework of quantum hypothesis testing. In many cases one can explicitly find the optimal measurement in the limit where one has simultaneous access to a large number $n$ of identical copies of the system, and estimate the expected error as $n$ becomes large. Interestingly, error estimates turn out to involve various quantum information theoretic quantities such as relative entropy, thereby giving these quantities operational meaning. In this paper we consider the application of quantum hypothesis testing to quantum many-body systems and quantum field theory. We review some of the necessary background material, and study in some detail the situation where the two states one wants to distinguish are parametrically close. The relevant error estimates involve quantities such as the variance of relative entropy, for which we prove a new inequality. We explore the optimal measurement strategy for spin chains and two-dimensional conformal field theory, focusing on the task of distinguishing reduced density matrices of subsystems. The optimal strategy turns out to be somewhat cumbersome to implement in practice, and we discuss a possible alternative strategy and the corresponding errors.
Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.
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