相关论文: Trace formula in noncommutative geometry and the z…
We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the…
We try to define the sequence of zeros of the Riemann zeta function by an intrinsic property. Let $(z_k)_{k\in \mathbb{N}}$ be the sequence of nontrivial zeros of $\zeta(s)$ with positive imaginary part. We write $z_k= 1/2+i\tau_k$ (RH says…
We prove the Riemann Hypothesis via an analytically regulated surface integral over the critical strip of the Riemann zeta function. The key idea is that the convergence of this normalized integral is equivalent to the condition that all…
We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find…
We show that formulas differing from classical analogues of rational trace formulas for algebraic-geometric potentials occur in the theory of finite-gap integration of spectral equations. The new formulas contain transcendental modular…
In this paper we study the metric geometry of the space $\Sigma$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $\tau$. The trace induces an incomplete Riemannian…
We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…
The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical M\"untz formula [4]. These formulas are derived in the critical strip and in the half-plane…
Assuming the Riemann hypothesis, we obtain asymptotic formulas for $\sum_{0<\gamma<T}\zeta(\rho+\delta)\zeta(1-\rho+\overline{\delta})$ in the region $-\frac{a}{\log T} \leq \Re \delta \leq \frac{1}{2}+\frac{a}{\log T}$, $|\Im \delta|\ll…
An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that…
The introduction of strings into the study of the Riemann Hypothesis provides a visualization of the genesis of zeros for the Zeta function. The method is heuristic and when originally introduced suggested strong visual evidence for the…
The continuous spectrum to the spectral side of the Arthur-Selberg trace formula is described in terms of intertwining operators, whose normalising factors involve quotients of $L$-functions. In this paper, we derive two expressions in the…
We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…
It is well known that the distribution of the prime numbers plays a central role in number theory. It has been known, since Riemann's memoir in 1860, that the distribution of prime numbers can be described by the zero-free region of the…
We give an interpretation of the Riemann hypothesis in terms of complex and topological dynamics. For example, the Riemann hypothesis is affirmative and all zeros of the Riemann zeta function are simple if and only if a certain meromorphic…
Finding hidden order within disorder is a common interest in material science, wave physics, and mathematics. The Riemann hypothesis, stating the locations of nontrivial zeros of the Riemann zeta function, tentatively characterizes…
In 2016, the first-named author introduced a formulation of the Alternative Hypothesis that assumes that consecutive zeros of the Riemann zeta-function are spaced at multiples of half of the average spacing, but does not assume that the…
This paper presents a combinatorial study of sums of integer powers of the cotangent which is a popular theme in classical calculus. Our main tool the realization of cotangent values as eigenvalues of a simple self-adjoint matrix with…
The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…
Global mapping properties of the Riemann Zeta function are used to investigate its non trivial zeros.