Related papers: Ruelle zeta function at zero for surfaces

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good…

We propose a field-theoretic interpretation of Ruelle zeta function, and show how it can be seen as the partition function for $BF$ theory when an unusual gauge fixing condition on contact manifolds is imposed. This suggests an alternative…

Space-like maximal surfaces and time-like minimal surfaces in Lorentz-Minkowski3-space are both characterized as zero mean curvature surfaces. We are interested in the case where the zero mean curvature surface changes type from space-like…

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.

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.

Principal value integrals are associated to multi-valued rational differential forms with normal crossings support on a non-singular algebraic variety. We prove their vanishing on rational surfaces in the context of a conjecture of…

We prove two rigidity results for surfaces lying in the standard null hypersurfaces of Schwarzschild spacetime satisfying certain mean curvature type equations. The first is for the equation $\alpha_H = - d\log |H|$ studied in \cite{WWZ}.…

We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow…

Let $p$ and $l$ be rational primes such that $l$ is odd and the order of $p$ modulo $l$ is even. For such primes $p$ and $l$, and for $e=l, 2l$, we consider the non-singular projective curves $aY^e = bX^e + cZ^e$ ($abc \neq 0$) defined over…

We will prove that the zeta function for Ruelle-expanding maps is rational.

The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is…

We prove that for manifolds with negative curvature bounded away from $0$ of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes…

By considering the prime zeta function, the author intended to demonstrate in that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which would have proven the Riemann hypothesis. However, he later realised that the proof of…

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s=…

An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no…

We establish a principal value integral formula, for the residue of the largest non-trivial candidate pole of the real or complex local zeta function associated to an analytic germ f, which is non-degenerate with respect to its Newton…

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic $p > 5$. As a consequence, we deduce the Grauert-Riemenschneider vanishing theorem for excellent divisorial log terminal…

We derive several vanishing theorems for genera under almost nonnegative Ricci curvature and infinite fundamental group, which includes Todd genus, $\widehat{A}$-genus, elliptic genera and Witten genus. A vanishing theorem of Euler…

We establish Noether's inequality for surfaces of general type in positive characteristic.Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all…

Recently a function was constructed that satisfies all known properties of a tree-level scattering of four massless scalars via the exchange of an infinite tower of particles with masses given by the non-trivial zeroes of the Riemann zeta…