Riemann Hypothesis Holds virtually true

If infinity times zero equal zero then Zeta function has an analytic continuity over the whole complex plan except a simple pole at 1. Since infinity times zero is undefined, then Riemann' s approach remain not sharp. However, it is true that the non trivial zeros lie at the critical line x = 1/2 despite there simultaneous virtual existence or in another words, such zeros are assumed to exist.

A criterion for univalent meromorphic functions

Let D = {z ? C,|z| < 1} and A(p) be the set of meromorphic functions in D possessing only simple pole at the point p with p ? (0,1). The aim of this paper is to give a criterion by mean of conditions on the parameters ?,? ? C, ? > 0 and g ? A(p) for functions in the class denoted P ?,?,h(p; ?) of functions f ? A(p) satisfying a differential Inequality of the form |?(z/f(z))" + ?(z/g(z))"|? ??, z ? D to be univalent in the disc D, where ? = (1-p/1+p)2.

Generating functions on covering groups

In this paper we prove a conjecture relating the Whittaker function of a certain generating function with the Whittaker function of the theta representation $\unicode[STIX]{x1D6E9}_{n}^{(n)}$. This enables us to establish that a certain global integral is factorizable and hence deduce the meromorphic continuation of the standard partial $L$ function $L^{S}(s,\unicode[STIX]{x1D70B}^{(n)})$. In fact we prove that this partial $L$ function has at most a simple pole at $s=1$. Here, $\unicode[STIX]{x1D70B}^{(n)}$ is a genuine irreducible cuspidal representation of the group $\text{GL}_{r}^{(n)}(\mathbf{A})$.

SPECTRAL ZETA FUNCTIONS FOR THE QUANTUM RABI MODELS

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at $s=1$. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

A Numerical Test of Padé Approximation for Some Functions with Singularity

The aim of this study is to examine some numerical tests of Padé approximation for some typical functions with singularities such as simple pole, essential singularity, brunch cut, and natural boundary. As pointed out by Baker, it was shown that the simple pole and the essential singularity can be characterized by the poles of the Padé approximation. However, it was not fully clear how the Padé approximation works for the functions with the branch cut or the natural boundary. In the present paper, it is shown that the poles and zeros of the Padé approximated functions are alternately lined along the branch cut if the test function has branch cut, and poles are also distributed around the natural boundary for some lacunary power series and random power series which rigorously have a natural boundary on the unit circle. On the other hand, Froissart doublets due to numerical errors and/or external noise also appear around the unit circle in the Padé approximation. It is also shown that the residue calculus for the Padé approximated functions can be used to confirm the numerical accuracy of the Padé approximation and quasianalyticity of the random power series.