Related papers: Hilbert-Polya conjecture and Generalized Riemann H…

Let p be an odd prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L-function which interpolates the complex…

Y\i ld\i r\i m has classified zeros of the derivatives of Dirichlet $L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this paper we remove the possibility of vagrant zeros for $L'(s,\chi)$ when the conductors are…

Using the method of multiple Dirichlet series, we develop L-functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for quadratic families of Dirichlet and Hecke L-functions of primerelated moduli…

There exist small perturbations of L-functions, satisfying the appropriate functional equation, for which the analogue of the Riemann hypothesis fails radically. Moreover, this phenomenon is generic. However, there also exist small…

Towards the Hilbert-P\'olya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes…

We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery's function $F(\alpha, T)$ in bounded intervals, the second is an integral introduced by Selberg…

We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to…

Recently, we have established the generalized Li's criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_rho(1-(1-((rho-a)/(rho+a-1))^n) for any real a not…

In this paper, we prove the Generalized P\'{o}lya conjecture for the Dirichlet eigenvalues. In other words, we show that $\lambda_k(\alpha) \ge \frac{(2\pi)^{\alpha} k^{\alpha/n}}{\big(\omega_n \cdot {vol}(\Omega)\big)^{\alpha/n}}, \quad\,…

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= \lambda h(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a…

The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its…

Recently the author has presented a new approach to solving extremal problems of geometric function theory. It involves the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces. We show here that this approach,…

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…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

The celebrated Riemann-Siegel formula compares the Riemann zeta function on the critical line with its partial sums, expressing the difference between them as an expansion in terms of decreasing powers of the imaginary variable $t$. Siegel…

We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large…

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$, $$ \cL^{(\alpha)}_{\sigma,b} f(x):=\mbox{p.v.} \int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\dif z+b(x)\cdot\nabla f(x)+\sL f(x), $$ where…

A generalized modular relation of the form $F(z, w, \alpha)=F(z, iw,\beta)$, where $\alpha\beta=1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann $\Xi$-function. It is a two-variable…

We use expansions with functions related to some special functions such as Hermite or Laguerre to get some conjectural expansions of the Riemann Zeta function in the critical strip involving a set of polynomials which have their zeros on…

In this paper, some new results are reported for the study of Riemann zeta function $\zeta(s)$ in the critical strip $0<Re(s)<1$, such as $\zeta(s)$ expressed in a generalized Euler product only involving prime numbers. Particularly, some…