Selberg class

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.^[1]

Definition

The formal definition of the class S is the set of all Dirichlet series

F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}

absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them):

Analyticity: $F(s)$ has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s equals 1.
Ramanujan conjecture: a₁ = 1 and $a_{n}\ll _{\varepsilon }n^{\varepsilon }$ for any ε > 0;
Functional equation: there is a gamma factor of the form
$\gamma (s)=Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})$

where Q is real and positive, Γ the gamma function, the ω_i real and positive, and the μ_i complex with non-negative real part, as well as a so-called root number

$\alpha \in \mathbb {C} ,\;|\alpha |=1$ ,

such that the function

$\Phi (s)=\gamma (s)F(s)\,$

satisfies

$\Phi (s)=\alpha \,{\overline {\Phi (1-{\overline {s}})}};$
Euler product: For Re(s) > 1, F(s) can be written as a product over primes:
$F(s)=\prod _{p}F_{p}(s)$

with

$F_{p}(s)=\exp \left(\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}\right)$

and, for some θ < 1/2,

$b_{p^{n}}=O(p^{n\theta }).\,$

Comments on definition

The condition that the real part of μ_i be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μ_i 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 case includes $(1-2^{-s})(1-2^{1-s})$ whose zeros are not on the critical line.

Without the condition $a_{n}\ll _{\varepsilon }n^{\varepsilon }$ there would be $L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})$ which violates the Riemann hypothesis.

It is a consequence of 4. that the a_n are multiplicative and that

F_{p}(s)=\sum _{n=0}^{\infty }{\frac {a_{p^{n}}}{p^{ns}}}{\text{ for Re}}(s)>0.

Examples

The prototypical example of an element in S is the Riemann zeta function.^[2] Another example, is the L-function of the modular discriminant Δ

L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}

where $a_{n}=\tau (n)/n^{11/2}$ and τ(n) is the Ramanujan tau function.^[3]

All known examples are automorphic L-functions, and the reciprocals of F_p(s) are polynomials in p^−s of bounded degree.^[4]

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.^[5]

Basic properties

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 N_F(T),^[6] Selberg showed that

N_{F}(T)=d_{F}{\frac {T\log(T+C)}{2\pi }}+O(\log T).

Here, d_F is called the degree (or dimension) of F. It is given by^[7]

d_{F}=2\sum _{i=1}^{k}\omega _{i}.

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

d_{FG}=d_{F}+d_{G}.

A function F ≠ 1 in S is called primitive if whenever it is written as F = F₁F₂, with F_i in S, then F = F₁ or F = F₂. If d_F = 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.^[8]

Selberg's conjectures

In (Selberg 1992), Selberg made conjectures concerning the functions in S:

Conjecture 1: For all F in S, there is an integer n_F such that $\sum _{p\leq x}{\frac {|a_{p}|^{2}}{p}}=n_{F}\log \log x+O(1)$ and n_F = 1 whenever F is primitive.
Conjecture 2: For distinct primitive F, F′ ∈ S, $\sum _{p\leq x}{\frac {a_{p}{\overline {a_{p}^{\prime }}}}{p}}=O(1).$
Conjecture 3: If F is in S with primitive factorization $F=\prod _{i=1}^{m}F_{i},$ χ is a primitive Dirichlet character, and the function $F^{\chi }(s)=\sum _{n=1}^{\infty }{\frac {\chi (n)a_{n}}{n^{s}}}$ is also in S, then the functions F_i^χ are primitive elements of S (and consequently, they form the primitive factorization of F^χ).
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

Conjectures 1 and 2 imply that if F has a pole of order m at s = 1, then F(s)/ζ(s)^m is entire. In particular, they imply Dedekind's conjecture.^[9]

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.^[10]

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 n_F = 1.^[11]

Notes

^ 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.
^ Murty 2008
^ Murty 2008
^ Murty 1994
^ Jerzy Kaczorowski & Alberto Perelli (2011). "On the structure of the Selberg class, VII" (PDF). Annals of Mathematics. 173: 1397–1441. doi:10.4007/annals.2011.173.3.4.
^ The zeroes on the boundary are counted with half-multiplicity.
^ While the ω_i 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 $F$ of $\mathbb {Q}$ , the zeta function $\zeta _{F}(s)$ is divisible by the Riemann zeta function $\zeta (s)$ . That is, the quotient $\zeta _{F}(s)/\zeta (s)$ is entire. More generally, Dedekind conjectures that if $K$ is a finite extension of $F$ , then $\zeta _{K}(s)/\zeta _{F}(s)$ should be entire. This conjecture is still open.
^ Murty 1994, Theorem 4.3
^ Conrey & Ghosh 1993, § 4

References

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, 31 (1): 1–14, arXiv:math/9407219, doi:10.1090/s0273-0979-1994-00479-3, MR 1242382, S2CID 265909, Zbl 0805.11062

Murty, M. Ram (2008), Problems in analytic number theory, Graduate Texts in Mathematics, Readings in Mathematics, vol. 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

[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.

[2] Murty 2008

[3] Murty 2008

[4] Murty 1994

[5] Jerzy Kaczorowski & Alberto Perelli (2011). "On the structure of the Selberg class, VII" (PDF). Annals of Mathematics. 173: 1397–1441. doi:10.4007/annals.2011.173.3.4.

[6] The zeroes on the boundary are counted with half-multiplicity.

[7] While the ω_i are not uniquely defined by F, Selberg's result shows that their sum is well-defined.

[8] Murty 1994, Lemma 4.2

[9] A celebrated conjecture of Dedekind asserts that for any finite algebraic extension $F$ of $\mathbb {Q}$ , the zeta function $\zeta _{F}(s)$ is divisible by the Riemann zeta function $\zeta (s)$ . That is, the quotient $\zeta _{F}(s)/\zeta (s)$ is entire. More generally, Dedekind conjectures that if $K$ is a finite extension of $F$ , then $\zeta _{K}(s)/\zeta _{F}(s)$ should be entire. This conjecture is still open.

[10] Murty 1994, Theorem 4.3

[11] Conrey & Ghosh 1993, § 4

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