Selberg class

Atle Selberg

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.

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):

1. Analyticity: $F(s)$ has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s = 1.
2. Ramanujan conjecture: a₁ = 1 and $a_{n}\ll _{\varepsilon }n^{\varepsilon }$ for any ε > 0;
3. 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

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

4. 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

• 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.

• The functional equation does not have to be unique. By the duplication formula for the $\Gamma$ function, new factors with different real constants can be produced. However, Selberg proved that the sum ${\textstyle \sum _{i=1}^{k}\omega _{i}}$ is independent of the choice of functional equation.

• 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–Petersson conjecture holds, and which 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.

Properties

The Selberg class is closed under multiplication of functions: if F and G are in the Selberg class, then so is their product.

From the Ramanujan conjecture, it follows that, for every $\epsilon >0$, ${\textstyle \sum _{i=1}^{n}\vert a_{i}\vert =O(n^{1+\epsilon })}$; hence, the Dirichlet series is absolutely convergent in the half-plane ${\textstyle {\text{Re}}(s)>1}$.

Despite the unusual version of the Euler product in the axioms, by exponentiation of Dirichlet series, one can deduce that a_n is a multiplicative sequence and that

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

The real non-negative number

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

is called the degree (or dimension) of F. Since this sum is independent of the choice of functional equation, it is well-defined for any function F. If F and G are in the Selberg class, then so is their product, and

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

It can be shown that F = 1 is the only function in S whose degree is ${\textstyle d_{F}<1}$. Kaczorowski and Perelli showed that the only cases of ${\textstyle d_{F}<2}$ are the Dirichlet L-functions for primitive Dirichlet characters (including the Riemann zeta-function).

From the Euler product and Ramanujan conjecture, it follows that, for ${\textstyle {\text{Re}}(s)>1}$, functions in Selberg class are non-vanishing. From the functional equation, every pole of the gamma factor γ(s) in ${\textstyle {\text{Re}}(s)<0}$ must be cancelled by a zero of F. Such zeroes are called trivial zeroes; the other zeroes of F are called non-trivial zeroes. All nontrivial zeroes are located in the critical strip, ${\textstyle 0<{\text{Re}}(s)<1}$, and by the functional equation, the nontrivial zeroes are symmetrical about the critical line, ${\textstyle {\text{Re}}(s)={\frac {1}{2}}}$. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by N_F(T), Selberg showed that:

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

An explicit version of the result was proven by Palojärvi.

It was proven by Kaczorowski & Perelli that, for F in the Selberg class, ${\textstyle F(1+it)\neq 0}$ for $t\in \mathbb {R}$ is equivalent to

$\lim _{x\rightarrow \infty }{\frac {\sum _{p\leq x}\vert a_{p}\vert ^{2}}{\pi (x)}}=\kappa _{F},$

where $\kappa _{F}>0$ is a real number and ${\textstyle \pi }$ is the prime-counting function. This result can be thought of as a generalization of the prime number theorem.

Nagoshi & Steuding showed that functions satisfying the prime-number theorem condition have a universality property for the strip ${\textstyle \sigma <Re(s)<1}$, where ${\textstyle \sigma =\max \lbrace {\frac {1}{2}},1-{\frac {1}{d_{F}}}\rbrace }$. It generalizes the universality property of the Riemann zeta function and Dirichlet L-functions.

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; however, the uniqueness of such a factorization is still open problem.

Examples

The prototypical example of an element in S is the Riemann zeta function. Also, most of generalizations of the zeta function, like Dirichlet L-functions or Dedekind zeta functions, belong to the Selberg class.

Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters or Artin L-functions for irreducible representations.

Another example is the L-function of the modular discriminant Δ,

$L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {\tau (n)/n^{11/2}}{n^{s}}},$

where ${\textstyle \tau (n)}$ is the Ramanujan tau function. This example can be considered a "normalized" or "shifted" L-function for the original Ramanujan L-function, defined as

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

whose coefficients satisfy ${\textstyle \vert \tau (n)\vert \leq n^{\frac {11}{2}}}$. It has the functional equation

$({\frac {1}{2\pi }})^{s}\Gamma (s)L(s)=({\frac {1}{2\pi }})^{12-s}\Gamma (12-s)L(12-s)$

and is expected to have all nontrivial zeroes on the line ${\textstyle Re(s)=6}$.

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

Conjectures

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).
• Generalized Riemann hypothesis for S: For all F in S, the non-trivial zeroes of F all lie on the line Re(s) = 1/2.

The first two Selberg conjectures are often collectively called the Selberg orthogonality conjecture.

Other conjectures

It is conjectured that Selberg class is equal to class of automorphic L-functions.

It is conjectured that all reciprocals of factors F_p(s) of the Euler products are polynomials in p^−s of bounded degree.

It is conjectured that, for any F in the Selberg class, ${\textstyle d_{F}}$ is a nonnegative integer. The best particular result due to Kaczorowski & Perelli shows this only for ${\textstyle d_{F}<2}$.

Consequences of the conjectures

The Selberg orthogonality conjecture has numerous consequences for functions in the Selberg class:

• The factorization of function F in S into primitive functions would be unique.
• If ${\textstyle F=F_{1}^{e_{1}}\dots F_{n}^{e_{n}}}$ is a factorization of F in S into primitive functions, then ${\textstyle n_{F}=e_{1}^{2}+\dots +e_{n}^{2}}$. In particular, this implies that ${\textstyle n_{F}=1}$ if and only if F is a primitive function.
• The functions in S have no zeroes on ${\textstyle Re(s)=1}$. This implies that they satisfy a generalization of the prime number theorem and have a universality property.
• 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.
• M. Ram Murty showed in (Murty 1994) that the orthogonality conjecture implies the Artin conjecture.
• L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive.

The Generalized Riemann Hypothesis for S implies many different generalizations of the original Riemann Hypothesis, the most notable being the generalized Riemann hypothesis for Dirichlet L-functions and extended Riemann Hypothesis for Dedekind zeta functions, with multiple consequences in analytic number theory, algebraic number theory, class field theory, and numerous branches of mathematics.

Combined with the Generalized Riemann hypothesis, different versions of orthogonality conjecture imply certain growth rates for the function and its logarithmic derivative.

If the Selberg class equals the class of automorphic L-functions, then the Riemann hypothesis for S would be equivalent to the Grand Riemann hypothesis.

See also

• List of zeta functions

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