In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if n = P 1?(1?1==) n has n > 0 for n = 1; 2; 3; : : : where runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers and therefore is not speciic to zeta functions. We also give an a...

We consider open circular billiards with one and with two holes. The exact formulas for escape are obtained which involve the Riemann zeta function and Dirichlet L functions. It is shown that the problem of finding the exact asymptotics in the small hole limit for escape in some of these billiards is equivalent to the Riemann hypothesis. Escape from a circle and Riemann hypotheses 2

We continue our study of quantum Riemann surfaces initiated in 1, 2, 3]. We construct a one parameter family of deformations of compact Riemann surfaces of genus g 2. Our construction does not require any discretness condition on the value of Planck's constant. It coincides with the construction of 2] in the case when Planck's constant assumes the discrete set of values dictated by geometric qu...

In this article, we survey the asymptotic stability analysis of fractional differential systems with the Prabhakar fractional derivatives. We present the stability regions for these types of fractional differential systems. A brief comparison with the stability aspects of fractional differential systems in the sense of Riemann-Liouville fractional derivatives is also given. 

Fej'{e}r  Hadamard  inequality is generalization of Hadamard inequality. In this paper we prove certain Fej'{e}r  Hadamard  inequalities for $k$-fractional integrals. We deduce Fej'{e}r  Hadamard-type  inequalities for Riemann-Liouville fractional integrals. Also as special case Hadamard inequalities for $k$-fractional as well as fractional integrals are given.

In this paper, the Schwarz boundary value problem (BVP) for the inhomogeneous Cauchy-Riemann equation in a triangle is investigated explicitly. Firstly, by the technique of parquetingreflection and the Cauchy-Pompeiu representation formula a modified Cauchy-Schwarz representation formula is obtained. Then, the solution of the Schwarz BVP is explicitly solved. In particular, the boundary behavio...

In this paper, by using the Euler-Maclaurin expansion for the Riemann-$zeta$ function, we establish an inequality of a weight coefficient. Using this inequality, we derive a new reverse Hilbert's type inequality. As an applications, an equivalent form is obtained.

In this article, we develop the distributed order fractional hybrid differential equations (DOFHDEs) with linear perturbations involving the fractional Riemann-Liouville derivative of order $0 < q < 1$ with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed $varphi$-Lipschit...

On a closed Riemann surface Ra of genus g there exist g linearly independent differentials of the first kind, wu • • • , ws,.and their integrals around 2g canonical cycles or retrosections, Oi, • • • , a„, bi, ••• , ba, are usually put together to form a gX2g matrix (an; ßn) = (A; B), i, j =1, • • • , g, where a,-,is the integral of w{ over a,. Riemann showed that the entries in this matrix wer...