On common zeros of L-functions

We will prove that two L-functions L ₁, L ₂ satisfying the same functional equation must be identically equal if sufficiently many zeros of L ₁ are also zeros of L ₂.     

Correlation of zeros of automorphic L-functions

We compute the n-level correlation of normalized nontrivial zeros of a product of L-functions:L(s,π1)···L(s,πk), where πj, j=1,...,k, are automorphic cuspidal representations of GLmj(QA). Here the sizes of the groups GLmj(QA) are not necessarily the same. When these L(s,πj) are distinct, we prove that their nontrivial zeros are uncorrelated, as predicted by random matrix theory and verified numerically. When L(s,πj) are not necessarily distinct, our results will lead to a proof that the n-level correlation of normalized nontrivial zeros of the product L-function follows the superposition of Gaussian Unitary Ensemble (GUE) models of individual L-functions and products of lower rank GUEs. The results are unconditional when m1,...,mk 4,but are under Hypothesis H in other cases.     

Inequalities for the zeros of dirichlet L-functions

Let χ denote a primitive, Dirichlet character to the modulus q>i and let L(s,χ) be the corresponding Dirichlet L-series defined by L(s,χ) = ∑χ(n)n^?s,s = σ+it, for σ>0. It is of interest to know where the zeros of L(s,χ) are located, since the location of these zeros would yield important results in number theory. In this paper, we show that the spectrum of each member of a certain class of Hermitian matrices leads to an explicit zero-free region for L(s,χ).     

Functoriality of automorphic L-functions through their zeros

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s,π1)···L(s,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), with that of L(s,π). We prove a necessary condition for L(s,π) having a factoriz...     

Real zeros of Eisenstein series and Rankin-Selberg L-functions

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $\tfrac{1}{5}$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.     

Mean value of L-functions and the number of classes of a cyclotomic field

The Paley-Selberg asymptotic formula is refined to the form

On special zeros of p-adic L-functions of Hilbert modular forms

Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the $\mathcal {L}$ -invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.     

Prime number theorems for Rankin-Selberg L-functions over number fields

In this paper we define a Rankin-Selberg L-function attached to automorphic cuspidal represen-tations of GLm(AE) × GLm (AF ) over cyclic algebraic number fields E and F which are invariant under the Galois action,by exploiting a result proved by Arthur and Clozel,and prove a prime number theorem for this L-function.     

On the zeros on the critical line of L-functions corresponding to automorphic cusp forms

We consider an automorphic cusp form of integer weight k ≥ 1, which is the eigenfunction of all Hecke operators. It is proved that, for the L-series whose coefficients correspond to the Fourier coefficients of such an automorphic form, the positive fraction of nontrivial zeros lie on the critical line.     

The distribution of the zeros of L-functions and the least prime in some arithmetic progression

This paper gives some new estimations to the distribution of the zeros of L-functions and proves that the least prime in an arithmetic progression with a prime differenceq is ≪q ^4.5.