Abstract. In 1952, Heinrich Scholz published a question in the Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached in terms of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally in terms of structural graph theory. Although Scholz' question was answered in various ways,

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The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second-order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing a direct logical characterization of rudimentary languages and a representation result of second-order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity.Mathematics Subject Classification: 03C13, 03D20.')The authors wish to express their thanks to E. GRANDJEAN for suggesting the problem and for 2)e-mail: Malika.MoreQmath.unicaen.fr 3)e-mai1: oiiveQlogique.jussieu.fr many stimulating conversations.

Ash's functions N σ,k count the number of k-equivalence classes of σ-structures of size n. Some conditions on their asymptotic behavior imply the long standing spectrum conjecture. We present a new condition which is equivalent to this conjecture and we discriminate some easy and difficult particular cases.

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