Related papers: Lorentz-Invariant Hamiltonian and Riemann Hypothes…

The Hamiltonian formulation for a non-Abelian gauge theory in two spatial dimensions is carried out in terms of a gauge-invariant matrix parametrization of the fields. The Jacobian for the relevant transformation of variables is given in…

The first nontrivial zeroes of the Riemann $\zeta$ function are $\approx 1/2+\pm14.13472i$. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so we try to quantify the notion that the…

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region.…

In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.

The $\theta=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions…

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…

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound…

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and…

In these lectures we first review the important properties of the Riemann $\zeta$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic…

We prove that if a function $\theta \left( z \right)=\int\limits_{1}^{\infty }{\frac{\pi \left( t \right)\,-Li\left( t \right)}{{{t}^{z+1}}}dt}\,,$ which is holomorphic in $\left\{ \operatorname{Re}z>1 \right\}$ holomorphically extends to…

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…

Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.

The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves.…

For the Riemann zeta-function, we introduce a function such that it is a characteristic function of an infinitely divisible distribution on the real line if and only if the Riemann Hypothesis is true.

The Riemann hypothesis, which states that the non-trivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. Inspired by the P\'olya-Hilbert…

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+i\lambda_n. A supersymmetric quantum mechanical model is proposed as an…

The Hamiltonian $H={1\over2} p^2+{1\over2}m^2x^2+gx^2(ix)^\delta$ with $\delta,g\geq0$ is non-Hermitian, but the energy levels are real and positive as a consequence of ${\cal PT}$ symmetry. The quantum mechanical theory described by $H$ is…

Chaos quantization conditions, which relate the eigenvalues of a Hermitian operator (the Riemann operator) with the non-trivial zeros of the Riemann zeta function are considered, and their geometrical interpretation is discussed.

This brief note explicates some mathematical details of Phys. Rev. Lett. 118, 130201 (2017), by showing how a version of the operator of that paper can be rigorously constructed on a well-defined linear space of functions.

We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill-Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of…