Alberto Perelli

On the structure of the Selberg class, I: 0≤d≤1

The Selberg class S is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global L-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of S. Such a classication is based on a real-valued invariant d called degree, and the degree conjecture asserts that d 2 N for every L-function in S. The degree conjecture has been proved for d < 5=3, and in this paper we extend its validity to d < 2. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the L-functions in S.

Goldbach numbers represented by polynomials.

Let N be a large positive real number. It is well known that almost all even integers in the interval [N, 2N] are Goldbach numbers, i.e. a sum of two primes. The same result also holds for short intervals of the form [N, N+H], see Mikawa [4], Perelli-Pintz [7] and Kaczorowski-Perelli-Pintz [3] for the choice of admissible values of H and the size of the exceptional set in several problems in this direction. One may ask if similar results hold for thinner sequences of integers in [N, 2N], of cardinality smaller than the upper bound for the exceptional set in the above problems. In this paper we deal with the polynomial case.

Linear independence of L-functions

Abstract We prove the linear independence of the L-functions, and of their derivatives of any order, in a large class 𝒞 defined axiomatically. Such a class contains in particular the Selberg class as well as the Artin and the automorphic L-functions. Moreover, 𝒞 is a multiplicative group, and hence our result also proves the linear independence of the inverses of such L-functions.

Strong multiplicity one for the Selberg class

Abstract Let S denote the Selberg class of L -functions. We prove the strong multiplicity one property for the subclass of functions F∈ S with polynomial Euler product.

A converse theorem for Dirichlet L-functions

It is known that the space of solutions (in a suitable class of Dirichlet series with continuation over C) of the functional equation of a DirichletL-functionL.s; / has dimension 2 as soon as the conductor q of is at least 4. Hence the Dirichlet L-functions are not characterized by their functional equation for q 4. Here we characterize the conductors q such that for every primitive character (mod q), L.s; / is the only solution with an Euler product in the above space. It turns out that such conductors are of the form q D 23m with any square-free m coprime to 6 and finitely many a and b. Mathematics Subject Classification (2000). 11M06, 11M41.

The addition of primes and power

Let k > 2 be an integer. Let Ek(N) be the number of natural numbers not exceeding N which are not the sum of a prime and a &-th power of a natural number. Assuming the Riemann Hypothesis for all Dirichlet L-functions it is shown that £ * ( # ) < N~M.

Some remarks on the convergence of the Dirichlet series of L-functions and related questions

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of L-functions from the Selberg class

A correspondence theorem for L-functions and partial differential operators

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