Selberg class

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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[edit]

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)mF(s) is an entire function of finite order for some non-negative integer m;
  2. Ramanujan conjecture: a1 = 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_is+\mu_i)
    where φ is real, Q real and positive, Γ is the gamma function, the ω1 real and positive, and the μi 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[edit]

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 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 θ < 1/2 is important, as the θ = 1/2 case includes the Dirichlet eta-function, which violates the Riemann hypothesis.[1]

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

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

Examples[edit]

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] Additionally, if F is in S and χ is a primitive Dirichlet character, then Fχ 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 Fp(s) are polynomials in ps of bounded degree.[4]

Basic properties[edit]

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),[5] Selberg showed that

N_F(T)=d_F\frac{T\log(T+C)}{2\pi}+O(\log T).

Here, dF is called the degree (or dimension) of F. It is given by[6]

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 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.[7]

Selberg's conjectures[edit]

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
\sum_{p\leq x}\frac{|a_p|^2}{p}=n_F\log\log x+O(1)
and nF = 1 whenever F is primitive.
  • Conjecture 2: For distinct primitive FF′ ∈ S,
\sum_{p\leq x}\frac{a_pa_p^\prime}{p}=O(1).
  • Conjecture 3: If
F=\prod_{i=1}^mF_i
is a factorization of F into primitive functions and χ is a primitive Dirichlet character, then
F^\chi=\prod_{i=1}^mF_i^\chi
and the Fiχ 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[edit]

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.[8]

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.[9]

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.[10]

See also[edit]

Notes[edit]

  1. ^ Conrey & Ghosh 1993, §1
  2. ^ Lapidus, Michel Laurent (2008). In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes. American Mathematical Society. p. 389. ISBN 0821842226. Zbl 1150.11003. 
  3. ^ Murty 2008
  4. ^ Murty 1994
  5. ^ The zeroes on the boundary are counted with half-multiplicity.
  6. ^ While the ωi are not uniquely defined by F, Selberg's result shows that their sum is well-defined.
  7. ^ Murty 1994, Lemma 4.2
  8. ^ 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.
  9. ^ Murty 1994, Theorem 4.3
  10. ^ Conrey & Ghosh 1993, § 4

References[edit]

  • 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)