Related papers: Zeros of some level 2 Eisenstein series

In this paper we consider the constant term $\phi_K(y,s)$ of the non-normalized Eisenstein series attached to $\PSL(2,\cO_K)$, where $K$ is either $\Q$ or an imaginary quadratic field of class number one. The main purpose of this paper is…

The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other…

Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has…

Assuming the existence of a sequence of exceptional discriminants of quadratic fields, we show that a hundred percent of zeros of the Riemann zeta function are on the critical line in specific segments. This is a special case of a more…

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…

For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards…

We show that the dilogarithm has at most one zero on each branch, that each zero is close to a root of unity, and that they may be found to any precision with Newton's method. This work is motivated by applications to the asymptotics of…

We introduce a new method to detect the zeros of the Riemann zeta function which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few `half-isolated' zeros. By combining this with classical methods,…

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new…

We generalize the construction from arXiv:2102.09329 of theta series for quadratic forms of signature $(n-1,1)$ with homogeneous and spherical polynomials. Namely, we allow that the parameters $c_1,c_2$, which define the theta series and…

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if f is a classical holomorphic modular…

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $\alpha>-1$, $-2<\beta<-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that…

We study the arithmetic of degree $N-1$ Eisenstein cohomology classes for locally symmetric spaces associated to $\mathrm{GL}_N$ over an imaginary quadratic field $k$. Under natural conditions we evaluate these classes on $(N-1)$-cycles…

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…

In this paper, we show that incoherent Hilbert Eisenstein series for a real quadratic fields can be expressed as the Doi-Naganums lift of an incoherent Eisenstein series over $\mathbb{Q}$. As an application, we show when $N$ is odd and…

Elliptic integrals, since Euler's finding of addition theorem 1751, has been studied extensively from various view points. Present paper gives a view point from primitive integrals of types $\mathrm{A_2}, \mathrm{B_2}$ and $\mathrm{G_2}$…

We study the low lying zeros of $GL(2) \times GL(2)$ Rankin-Selberg $L$-functions. Assuming the generalized Riemann hypothesis, we compute the $1$-level density of the low-lying zeroes of $L(s, f \otimes g)$ averaged over families of…

We consider a certain family of Kudla-Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1,1), and prove that the arithmetic degrees of these cycles can be identified with the Fourier…

In this article, we prove some normality criteria for a family of meromorphic functions having zeros with some multiplicity. Our main result involves sharing of a holomorphic function by certain differential polynomials. Our results…

In this paper, we compute for odd fundamental discriminants $D>1$ the Fourier expansion of non-holomorphic elliptic Eisenstein series for $\Gamma_0(D)$ with quadratic nebentypus character $\chi_D$ satisfying a certain plus space condition.…