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Register a new userProof of Riemann Hypothesis
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Jinzhu Han
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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.
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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.
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.
The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of $zeta(s)$ lie on the line $Re(s) =1/2$. The density hypothesis is a conjectured estimate $N(lambda, T) =Obigl(Tsp{2(1-lambda) +epsilon} bigr)$ for any $epsilon >0$, where $N(lambda, T)$ is the number of zeros of $zeta(s)$ when $Re(s) gelambda$ and $0 <Im(s) le T$, with $1/2 le lambda le 1$ and $T >0$. The Riemann-von Mangoldt Theorem confirms this estimate when $lambda =1/2$, with $Tsp{epsilon}$ being replaced by $log T$. In an attempt to transform Backlunds proof of the Riemann-von Mangoldt Theorem to a proof of the density hypothesis by convexity, we discovered a different approach utilizing an auxiliary function. The crucial point is that this function should be devised to be symmetric with respect to $Re(s) =1/2$ and about the size of the Euler Gamma function on the right hand side of the line $Re(s) =1/2$. Moreover, it should be analytic and without any zeros in the concerned region. We indeed found such a function, which we call pseudo-Gamma function. With its help, we are able to establish a proof of the density hypothesis. Actually, we give the result explicitly and our result is even stronger than the original density hypothesis, namely it yields $N(lambda, T) le 8.734 log T$ for any $1/2 < lambda < 1$ and $Tge 2445999554999$.
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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