In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions. Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to automorphic forms and the Riemann hypothesis. The class was defined by Atle Selberg in (Selberg 1992), who preferred not to use the word "axiom" that later authors have employed.
The formal definition of the class S is the set of all Dirichlet series
absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them):
- Analyticity: the function (s − 1)mF(s) is an entire function of finite order for some non-negative integer m;
- Ramanujan conjecture: a1 = 1 and for any ε > 0;
- Functional equation: there is a gamma factor of the form
where Q is real and positive, Γ the gamma function, the ωi real and positive, and the μi complex with non-negative real part, as well as a so-called root number
such that the function
- Euler product: For Re(s) > 1, F(s) can be written as a product over primes:
and, for some θ < 1/2,
Comments on definition
The condition that the real part of μi be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μi is negative. Specifically, there are Maass forms associated with exceptional eigenvalues, for which the Ramanujan–Peterssen conjecture holds, and have a functional equation, but do not satisfy the Riemann hypothesis.
The condition that θ < 1/2 is important, as the θ = 1/2 case includes the Dirichlet eta-function, which violates the Riemann hypothesis.
It is a consequence of 4. that the an are multiplicative and that
where and τ(n) is the Ramanujan tau function.
All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.
The best results on the structure of the Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet L-functions (including the Riemann zeta-function) are the only examples with degree less than 2.
As with the Riemann zeta function, an element F of S has trivial zeroes that arise from the poles of the gamma factor γ(s). The other zeroes are referred to as the non-trivial zeroes of F. These will all be located in some strip 1 − A ≤ Re(s) ≤ A. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by NF(T), Selberg showed that
- It can be shown that F = 1 is the only function in S whose degree is less than 1.
If F and G are in the Selberg class, then so is their product and
A function F ≠ 1 in S is called primitive if whenever it is written as F = F1F2, with Fi in S, then F = F1 or F = F2. If dF = 1, then F is primitive. Every function F ≠ 1 of S can be written as a product of primitive functions. Selberg's conjectures, described below, imply that the factorization into primitive functions is unique.
Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters. Assuming conjectures 1 and 2 below, L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive.
In (Selberg 1992), Selberg made conjectures concerning the functions in S:
- Conjecture 1: For all F in S, there is an integer nF such that
- and nF = 1 whenever F is primitive.
- Conjecture 2: For distinct primitive F, F′ ∈ S,
- Conjecture 3: If F is in S with primitive factorization
- χ is a primitive Dirichlet character, and the function
- is also in S, then the functions Fiχ are primitive elements of S (and consequently, they form the primitive factorization of Fχ).
- Riemann hypothesis for S: For all F in S, the non-trivial zeroes of F all lie on the line Re(s) = 1/2.
Consequences of the conjectures
M. Ram Murty showed in (Murty 1994) that conjectures 1 and 2 imply the Artin conjecture. In fact, Murty showed that Artin L-functions corresponding to irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures.
The functions in S also satisfy an analogue of the prime number theorem: F(s) has no zeroes on the line Re(s) = 1. As mentioned above, conjectures 1 and 2 imply the unique factorization of functions in S into primitive functions. Another consequence is that the primitivity of F is equivalent to nF = 1.
- The title of Selberg's paper is somewhat a spoof on Paul Erdős, who had many papers named (approximately) "(Some) Old and new problems and results about...". Indeed, the 1989 Amalfi conference was quite surprising in that both Selberg and Erdős were present, with the story being that Selberg did not know that Erdős was to attend.
- Conrey & Ghosh 1993, §1
- Murty 2008
- Murty 2008
- Murty 1994
- Jerzy Kaczorowski and Alberto Perelli (2011). "On the structure of the Selberg class, VII." (PDF). Annals of mathematics. 173. pp. 1397––1411. doi:10.4007/annals.2011.173.3.4.
- The zeroes on the boundary are counted with half-multiplicity.
- While the ωi are not uniquely defined by F, Selberg's result shows that their sum is well-defined.
- Murty 1994, Lemma 4.2
- A celebrated conjecture of Dedekind asserts that for any ﬁnite algebraic extension of , the zeta function is divisible by the Riemann zeta function . That is, the quotient is entire. More generally, Dedekind conjectures that if is a ﬁnite extension of , then should be entire. This conjecture is still open.
- Murty 1994, Theorem 4.3
- Conrey & Ghosh 1993, § 4
- Selberg, Atle (1992), "Old and new conjectures and results about a class of Dirichlet series", Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno: Univ. Salerno, pp. 367–385, MR 1220477, Zbl 0787.11037 Reprinted in Collected Papers, vol 2, Springer-Verlag, Berlin (1991)
- Conrey, J. Brian; Ghosh, Amit (1993), "On the Selberg class of Dirichlet series: small degrees", Duke Mathematical Journal, 72 (3): 673–693, arXiv:math.NT/9204217, doi:10.1215/s0012-7094-93-07225-0, MR 1253620, Zbl 0796.11037
- Murty, M. Ram (1994), "Selberg's conjectures and Artin L-functions", Bulletin of the American Mathematical Society, New Series, American Mathematical Society, 31 (1): 1–14, arXiv:math/9407219, doi:10.1090/s0273-0979-1994-00479-3, MR 1242382, Zbl 0805.11062
- Murty, M. Ram (2008), Problems in analytic number theory, Graduate Texts in Mathematics, Readings in Mathematics, 206 (Second ed.), Springer-Verlag, Chapter 8, doi:10.1007/978-0-387-72350-1, ISBN 978-0-387-72349-5, MR 2376618, Zbl 1190.11001