Related papers: On The Complex Zeros of The Riemann Zeta Function

Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory…

Starting from the classical integral representation of the $\zeta(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we…

In this paper is stablished a characterization of the solutions of the equation: zeta(z) = 0. Then such a characterization is used to give a proof for Riemann is Conjecture.

The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…

The Riemann Hypothesis, originally proposed by the eminent mathematician Bernard Riemann in 1859, remains one of the most profound challenges in number theory. It posits that all non-trivial zeros of the Riemann zeta function {\zeta}(s) are…

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying…

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…

This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis.…

In this paper we provide a proof of the Riemann Hypothesis by relating the non-trivial zeros of the zeta function to a certain Sturm-Liouville eigenvalue problem on a finite interval.

An incomplete Riemann zeta function can be expressed as a lower-bounded, improper Riemann-Liouville fractional integral, which, when evaluated at $0$, is equivalent to the complete Riemann zeta function. Solutions to Landau's problem with…

A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…

We develop a finite-dimensional, symmetric matrix framework associated with the Riemann zeta function for complex arguments s with Real(s) unequal 1/2.

We first construct a dynamical systems model which in its steady-state serves as an analytic continuation of the completed Riemann zeta function over the entire critical strip. The resulting mathematical construct involves a linear…

We have dealt with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and showed some evidence to indicate the hypothesis. We briefly…

The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the…

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…

The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In…

The Riemann Hypothesis states that the Riemann zeta function $\zeta(z)$ admits a set of ``non-trivial'' zeros that are complex numbers supposed to have real part $1/2$. Their distribution on the complex plane is thought to be the key to…

We study the relationship between the zeros of the Riemann zeta function and physical systems exhibiting supersymmetry, $PT$ symmetry and $SU(2)$ group symmetry. Our findings demonstrate that unbroken supersymmetry is associated with the…

For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial…