Jerzy Kaczorowski

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.

Oscillations of a given size of some arithmetic error terms

A general method of estimating the number of oscillations of a given size of arithmetic error terms is developed. Special attention is paid to the remainder terms in the prime number formula, in the Dirichlet prime number theorem for primes in arithmetic progressions and to the remainder term in the asymptotic formula for the number of square free divisors of an integer.

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.

O-estimates for a class of arithmetic error terms

The main aim of this paper is to present a general method of proving O-estimates for a class of arithmetic error terms. We assume that error terms in question are boundary values of harmonic functions on the upper half-plane satisfying certain subsidiary conditions. We prove a general theorem for an axiomatically defined class of such functions and then we show how this result can be used to give statements in concrete situations. As examples we treat the classical case of the remainder term in the prime number formula obtaining a new proof of the well-known result of J. E. Littlewood, and the case of the remainder term in the asymptotic formula for the summatory function of the square-free divisor function. In the latter case our result is new.

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.

ON THE SUM OF THE TWISTED EULER FUNCTION

We study an asymptotic formula for the sum of values of the Euler φ-function twisted by a real Dirichlet character. The error term is split into the arithmetic and the analytic part. The former is studied with minimal use of analytic tools in contrast to the latter, where the analysis depends heavily on the distribution of the non-trivial zeros of the corresponding Dirichlet L-function. The results of the present paper are an extension of a recent work by the authors, where the case of the classical Euler φ-function has been studied. The present, more general situation invites new technical difficulties. Not all of them can be successfully overcome. For instance, satisfactory omega results for the analytic part are proved in the case of an even Dirichlet character only. Nevertheless, a method providing good omega estimates for the arithmetic part as well as for the complete error term is developed. Moreover, it is noted that the Riemann Hypothesis for the involved Dirichlet L-function is equivalent to a sufficiently sharp estimation of the analytic part. This shows in particular that the arithmetic part can be much larger than the corresponding analytic part.

A correspondence theorem for L-functions and partial differential operators

Given an L-function F (s) from the extended Selberg class, we associate a function ΦF (x, y). We show that the functions ΦF (x, y) are, in the general case, the analogs of the modular forms associated with the GL2 L-functions. Indeed, we prove that every ΦF (x, y) is eigenfunction of a certain partial differential operator. Moreover, we prove a general correspondence theorem for such L-functions involving the functions ΦF (x, y). Let F (s) be a function in the extended Selberg class S]. This means that (s−1)mF (s) is entire of finite order for some non-negative integer m, F (s) is representable for σ > 1 as an absolutely convergent Dirichlet series with coefficients a(n) and satisfies the functional equation γ(s)F (s) = ωγ̄(1− s)F̄ (1− s) with |ω| = 1 and γ(s) = Q r ∏

Lower estimates related to the twisted Dedekind function

Let χ(modq) be a Dirichlet character. The main goal of this paper is to study oscillations of the difference E(x,χ) =∑n≤xψχ(n) −1 2 L(2,χ) L(4,χ2)x2, where ψχ(n) = n∏ p|n(1 + χ(p)/p) denotes the twisted Dedekind function. We prove that for infinitely many odd characters χ called “good”, we have E(x,χ) = Ω(xloglog x), and E(x,χ) = Ω±(xloglog x) when χ is real. We give a necessary and sufficient condition for χ to be good, and in particular we prove that all odd primitive characters are good. We show also that there are infinitely many moduli q ≥ 3, including all prime powers q ≥ 3, for which all odd characters χ(modq) are good.

Bounded mean oscillation and the distribution of primes

As usual, let π( x ) denote the number of prime numbers [les ] x and ψ( x ) the well known Chebyshevs function. Let E ( x ) denote either (ψ( x )− x )/√ x or (π( x )−li x )/(√ x /log x ), x [ges ]2. The study of E occupies a central place in the theory of primes. A classical result of Littlewood [ 7 ] states that E ( x )=Ω ± (log log log x ) as x tends to infinity, showing in particular that E is unbounded. We expect rather erratic behaviour of E , but still one can wonder if it belongs to one of the classic function spaces [Xscr ], necessarily containing some unbounded functions. Let us extend definition of E ( x ) for x E ( x )=0. A natural question is if it belongs to BMO, the space of functions with bounded mean oscillation, see, e.g. [ 2 ]. A locally integrable function f on the real line belongs to BMO if there exists a constant C such that for every bounded interval I ⊂ R we have formula here with a suitable constant α I ∈ R . [mid ] I [mid ] denotes here the length of I . Without any loss in generality one can take formula here the average of f over I (cf. [ 2 ], chapter VI). BMO is important and intensely studied in the complex analysis. It is obvious that BMO is larger than the space of bounded (measurable) functions and thus it seems a natural candidate for [Xscr ]. E ∈BMO would mean that E behaves in a certain predictable way. Otherwise, we obtain another confirmation of the big irregularity in the distribution of primes.