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).

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:

1. Analyticity: the function $(s-1)^{m}F(s)$ is an entire function of finite order for some non-negative integer m;
2. Ramanujan conjecture: $a_{1}=1$ and $a_{n}\ll _{\epsilon }n^{\epsilon }$ for any ε > 0;
3. Functional equation: there is a gamma factor of the form

   $\gamma (s)=e^{i\phi }Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})$

   where φ is real, Q real and positive, Γ is the gamma function, the $\omega _{i}$ real and positive, and the $\mu _{i}$ complex with non-negative real part, so that the function

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

   satisfies

   $\Phi (s)={\overline {\Phi (1-{\overline {s}})}};$

4. Euler product: F(s) can be written as a product over primes:

   $F(s)=\prod _{p}F_{p}(s){\text{ for Re}}(s)>1\,$

   with

   $\log F_{p}(s)=\sum _{n=0}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}$

   and, for some θ < 1/2,

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

Comments on definition

The condition that the real part of $\mu _{i}$ be positive is because there are known L-functions that do not satisfy the Riemann hypothesis when $\mu _{i}$ is zero or negative. Specifically, there are Maass cusp 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 $\theta <1/2$ is important, as the $\theta =1/2$ case includes the Dirichlet eta-function, 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.

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. Additionally, if F is in S and χ is a primitive Dirichlet character, then F^χ defined by

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

is also in S.

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

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), 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

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

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,

$\sum _{p\leq x}{\frac {a_{p}a_{p}^{\prime }}{p}}=O(1).$

• Conjecture 3: If

$F=\prod _{i=1}^{m}F_{i}$

is a factorization of F into primitive functions and χ is a primitive Dirichlet character, then

$F^{\chi }=\prod _{i=1}^{m}F_{i}^{\chi }$

and the F_i^χ are primitive.

• 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) is entire. In particular, they imply Dedekind's conjecture.

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 n_F = 1.

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 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, MR 1253620

• Murty, M. Ram (1994), "Selberg's conjectures and Artin L-functions", Bulletin of the American Mathematical Society, New Series, 31 (1), American Mathematical Society: 1–14, arXiv:math/9407219, MR 1242382

• Chapter 8 of Murty, M. Ram (2008), Problems in analytic number theory, Graduate Texts in Mathematics, Readings in Mathematics, vol. 206 (Second ed.), Springer-Verlag, doi:10.1007/978-0-387-72350-1, ISBN 978-0-387-72349-5, MR 2376618