Giuseppe Molteni

Upper and lower bounds at s=1 for certain Dirichlet series with Euler product

Estimates of the form L (j) (s,A) ,j, DA R A in the range |s 1| 1/logRA for general L-functions, where RA is a parameter related to the functional equation of L(s,A), can be quite easily obtained if the Ramanujan hypothesis is assumed. We prove the same estimates when the L-functions have Euler product of polynomial type and the Ramanujan hypothesis is replaced by a much weaker assumption about the growth of certain elementary symmetrical functions. As a consequence, we obtain an upper bound of this type for every L(s, ), where is an automorphic cusp form on GL(d,AK). We employ these results to obtain Siegel-type lower bounds for twists by Dirichlet characters of the third symmetric power of a Maass form.

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.

Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH

We have recently proved several explicit versions of the prime ideal theorem under GRH. Here we further explore the method, in order to deduce its strongest consequence for the case where x diverges.

Explicit smoothed prime ideals theorems under GRH

Let ψK be the Chebyshev function of a number field K. Let ψ K (x) := ∫ x 0 ψK(t) dt and ψ (2) K (x) := 2 ∫ x 0 ψ (1) K (t) dt. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences |ψ K (x) − x 2 | and |ψ K (x) − x 3 |. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals. Math. Comp. 85(300), 1875–1899 (2016). Electronically published on October 6, 2015. DOI: http://dx.doi.org/10.1090/mcom3039

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.

Primes in explicit short intervals on RH

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

REPRESENTATION OF A 2-POWER AS SUM OF k 2-POWERS: THE ASYMPTOTIC BEHAVIOR

A k-representation of an integer l is a representation of l as sum of k powers of 2, where representations differing by the order are considered as distinct. Let be the maximum number of such representations for integers l whose binary representation has exactly σ non-zero digits. can be recovered from via an explicit formula, thus in some sense is the fundamental object. In this paper we prove that tends to a computable limit as k diverges. This result improves previous bounds which were obtained with purely combinatorial tools.

Zeros of Dedekind zeta functions under GRH

Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedeking zeta function having imaginary part in [T-a, T+a]. We also prove a bound for the multiplicity of the zeros.

Spectral properties for a new composition of a matrix and a complex representation

A way to compose a matrix P and a flnite dimensional representation ‰ of C via a map h into a new matrix P ⁄h ‰ is deflned. Several results about the spectrum, eigenvectors, kernel and rank of P ⁄h ‰ are proved.

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=Hb for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentFp. This implies that the equationFa=Gb with (a, b)=1 has the unique solutionF=Hb andG=Ha in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.