Selberg class (original) (raw)

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Axiomatic definition of a class of L-functions

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.[1]

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

F ( s ) = ∑ n = 1 ∞ a n n s {\displaystyle F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}} {\displaystyle 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 ) {\displaystyle F(s)} {\displaystyle F(s)} has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s = 1.
  2. Ramanujan conjecture: a 1 = 1 {\displaystyle a_{1}=1} {\displaystyle a_{1}=1} and a n = O ( n ε ) {\displaystyle a_{n}=O(n^{\varepsilon })} {\displaystyle a_{n}=O(n^{\varepsilon })} for any ε > 0;
  3. Functional equation: there is a gamma factor of the form
    γ ( s ) = Q s ∏ i = 1 k Γ ( ω i s + μ i ) {\displaystyle \gamma (s)=Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})} {\displaystyle \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
    α ∈ C , | α | = 1 {\displaystyle \alpha \in \mathbb {C} ,\;|\alpha |=1} {\displaystyle \alpha \in \mathbb {C} ,\;|\alpha |=1},
    such that the function
    Φ ( s ) = γ ( s ) F ( s ) {\displaystyle \Phi (s)=\gamma (s)F(s)\,} {\displaystyle \Phi (s)=\gamma (s)F(s)\,}
    satisfies
    Φ ( s ) = α Φ ( 1 − s ¯ ) ¯ ; {\displaystyle \Phi (s)=\alpha \,{\overline {\Phi (1-{\overline {s}})}};} {\displaystyle \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 ) = ∏ p F p ( s ) {\displaystyle F(s)=\prod _{p}F_{p}(s)} {\displaystyle F(s)=\prod _{p}F_{p}(s)}
    with
    F p ( s ) = exp ⁡ ( ∑ n = 1 ∞ b p n p n s ) {\displaystyle F_{p}(s)=\exp \left(\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}\right)} {\displaystyle 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 θ ) . {\displaystyle b_{p^{n}}=O(p^{n\theta }).\,} {\displaystyle b_{p^{n}}=O(p^{n\theta }).\,}

L ( s + 1 / 3 , χ 4 ) L ( s − 1 / 3 , χ 4 ) {\displaystyle L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})} {\displaystyle L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})}

which violates the Riemann hypothesis.

L ( s , χ ) = L ( s , χ ⋆ ) ∏ p | q ( 1 − χ ⋆ ( p ) p − s ) {\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-\chi ^{\star }(p)p^{-s}\right)} {\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-\chi ^{\star }(p)p^{-s}\right)}

Despite this function satisfies any other axiom, from that additional factors follows that it has infinitely many zeros on Re ⁡ ( s ) = 0 {\displaystyle \operatorname {Re} (s)=0} {\displaystyle \operatorname {Re} (s)=0}. They are not symmetric with respect to Re ⁡ ( s ) = 1 2 {\displaystyle \operatorname {Re} (s)={\tfrac {1}{2}}} {\displaystyle \operatorname {Re} (s)={\tfrac {1}{2}}} and without functional equation proposed by Selberg it is hard to distinguish between trivial and nontrivial zeros of this function.

γ 2 ( s ) = C ⋅ γ 1 ( s ) where: C ∈ R {\displaystyle \gamma _{2}(s)=C\cdot \gamma _{1}(s)\quad {\text{where: }}C\in \mathbb {R} } {\displaystyle \gamma _{2}(s)=C\cdot \gamma _{1}(s)\quad {\text{where: }}C\in \mathbb {R} }.

However, by the multiplication formula the same gamma factor can be expressed in many different ways, involving different number of gamma functions with different constants. Despite of this, Selberg proved that the sum ∑ i = 1 k ω i {\textstyle \sum _{i=1}^{k}\omega _{i}} {\textstyle \sum _{i=1}^{k}\omega _{i}} is independent of the choice of the gamma factor formula.

F ( s ) = 1 − α i 2 L ( s , χ 5 ) + 1 + α i 2 L ( s , χ ¯ 5 ) for: α = 10 − 5 − 2 5 − 1 {\displaystyle F(s)={\frac {1-\alpha i}{2}}L(s,\chi _{5})+{\frac {1+\alpha i}{2}}L(s,{\overline {\chi }}_{5})\quad {\text{for: }}\alpha ={\frac {{\sqrt {10-{\sqrt {5}}}}-2}{{\sqrt {5}}-1}}} {\displaystyle F(s)={\frac {1-\alpha i}{2}}L(s,\chi _{5})+{\frac {1+\alpha i}{2}}L(s,{\overline {\chi }}_{5})\quad {\text{for: }}\alpha ={\frac {{\sqrt {10-{\sqrt {5}}}}-2}{{\sqrt {5}}-1}}}

despite having analytic continuation, having periodic coefficients (thus satisfying Ramanujan conjecture) and satisfying functional equation have zeros lying outside critical line.

The Selberg class is closed under multiplication of functions: product of each two functions belonging to S are also in S. It is also easy to check that if F is in S, then function involved in functional equation:

F ( s ¯ ) ¯ = ∑ n = 1 ∞ a n ¯ n s {\displaystyle {\overline {F({\overline {s}})}}=\sum _{n=1}^{\infty }{\frac {\overline {a_{n}}}{n^{s}}}} {\displaystyle {\overline {F({\overline {s}})}}=\sum _{n=1}^{\infty }{\frac {\overline {a_{n}}}{n^{s}}}}

satisfies axioms and is also in S. If F is entire function in S, then F ( s + i t ) {\textstyle F(s+it)} {\textstyle F(s+it)} for t ∈ R {\textstyle t\in \mathbb {R} } {\textstyle t\in \mathbb {R} } is also in S.

From the Ramanujan conjecture, it follows that, for every ϵ > 0 {\displaystyle \epsilon >0} {\displaystyle \epsilon >0}:

∑ i = 1 n | a i | = O ( n 1 + ϵ ) {\textstyle \sum _{i=1}^{n}\vert a_{i}\vert =O(n^{1+\epsilon })} {\textstyle \sum _{i=1}^{n}\vert a_{i}\vert =O(n^{1+\epsilon })}

then Dirichlet series defining function is absolutely convergent in the half-plane: Re ⁡ ( s ) > 1 {\textstyle \operatorname {Re} (s)>1} {\textstyle \operatorname {Re} (s)>1}.

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

F p ( s ) = ∑ n = 0 ∞ a p n p n s for Re ( s ) > 1. {\displaystyle F_{p}(s)=\sum _{n=0}^{\infty }{\frac {a_{p^{n}}}{p^{ns}}}{\text{ for Re}}(s)>1.} {\displaystyle F_{p}(s)=\sum _{n=0}^{\infty }{\frac {a_{p^{n}}}{p^{ns}}}{\text{ for Re}}(s)>1.}

From θ < 1 2 {\textstyle \theta <{\tfrac {1}{2}}} {\textstyle \theta <{\tfrac {1}{2}}} follows that for each factor of Euler product:

log ⁡ F p = ∑ n = 1 ∞ b p n p n s {\displaystyle \log F_{p}=\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}} {\displaystyle \log F_{p}=\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}}

is absolutely convergent in Re ⁡ ( s ) > 1 2 {\textstyle \operatorname {Re} (s)>{\tfrac {1}{2}}} {\textstyle \operatorname {Re} (s)>{\tfrac {1}{2}}}. Then F p ( s ) {\displaystyle F_{p}(s)} {\displaystyle F_{p}(s)} is absolutely convergent and F p ( s ) ≠ 0 {\displaystyle F_{p}(s)\neq 0} {\displaystyle F_{p}(s)\neq 0} in this region. In half-plane of absolute convergence of original Dirichlet series function is absolutely convergent product of non-vanishing factors, then for functions in Selberg class F ( s ) ≠ 0 {\textstyle F(s)\neq 0} {\textstyle F(s)\neq 0} in Re ⁡ ( s ) > 1 {\textstyle \operatorname {Re} (s)>1} {\textstyle \operatorname {Re} (s)>1}.

From the functional equation follows that every pole of the gamma factor γ(s) in Re ( s ) < 0 {\textstyle {\text{Re}}(s)<0} {\textstyle {\text{Re}}(s)<0} must be cancelled by a zero of F. Such zeros are called trivial zeros; the other zeros of F are called non-trivial zeros. All nontrivial zeros are located in the critical strip, 0 < Re ( s ) < 1 {\textstyle 0<{\text{Re}}(s)<1} {\textstyle 0<{\text{Re}}(s)<1}, and by the functional equation, the nontrivial zeros are symmetrical about the critical line, Re ( s ) = 1 2 {\textstyle {\text{Re}}(s)={\frac {1}{2}}} {\textstyle {\text{Re}}(s)={\frac {1}{2}}}.

The real non-negative number

d F = 2 ∑ i = 1 k ω i {\displaystyle d_{F}=2\sum _{i=1}^{k}\omega _{i}} {\displaystyle 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 degree of their product is:

d F G = d F + d G . {\displaystyle d_{FG}=d_{F}+d_{G}.} {\displaystyle d_{FG}=d_{F}+d_{G}.}

It can be shown that F = 1 is the only function in S whose degree is d F < 1 {\textstyle d_{F}<1} {\textstyle d_{F}<1}. Kaczorowski & Perelli (2011) showed that the only cases of d F < 2 {\textstyle d_{F}<2} {\textstyle d_{F}<2} are the Dirichlet _L_-functions for primitive Dirichlet characters (including the Riemann zeta-function). Denoting the number of non-trivial zeros of F with 0 ≤ Im(s) ≤ T by NF(T),[2] Selberg showed that:

N F ( T ) = d F T log ⁡ ( T + C ) 2 π + O ( log ⁡ T ) . {\displaystyle N_{F}(T)=d_{F}{\frac {T\log(T+C)}{2\pi }}+O(\log T).} {\displaystyle 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 (2019).

It was proven by Kaczorowski & Perelli (2003) that for F in the Selberg class, F ( 1 + i t ) ≠ 0 {\textstyle F(1+it)\neq 0} {\textstyle F(1+it)\neq 0} for t ∈ R {\displaystyle t\in \mathbb {R} } {\displaystyle t\in \mathbb {R} } is equivalent to

lim x → ∞ ∑ p ≤ x | a p | 2 π ( x ) = κ F , {\displaystyle \lim _{x\rightarrow \infty }{\frac {\sum _{p\leq x}\vert a_{p}\vert ^{2}}{\pi (x)}}=\kappa _{F},} {\displaystyle \lim _{x\rightarrow \infty }{\frac {\sum _{p\leq x}\vert a_{p}\vert ^{2}}{\pi (x)}}=\kappa _{F},}

where κ F > 0 {\displaystyle \kappa _{F}>0} {\displaystyle \kappa _{F}>0} is a real number and π {\textstyle \pi } {\textstyle \pi } is the prime-counting function. This result can be thought of as a generalization of the prime number theorem. Nagoshi & Steuding (2010) showed that functions satisfying the prime-number theorem condition have a universality property for the strip σ < R e ( s ) < 1 {\textstyle \sigma <Re(s)<1} {\textstyle \sigma <Re(s)<1}, where σ = max { 1 2 , 1 − 1 d F } {\textstyle \sigma =\max \lbrace {\frac {1}{2}},1-{\frac {1}{d_{F}}}\rbrace } {\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 {\displaystyle F\neq 1} {\displaystyle F\neq 1} in S is called primitive if, whenever it is written as F = F 1 ⋅ F 2 {\displaystyle F=F_{1}\cdot F_{2}} {\displaystyle F=F_{1}\cdot F_{2}}, with both of function in Selberg class, then F = F 1 {\displaystyle F=F_{1}} {\displaystyle F=F_{1}} or F = F 1 {\displaystyle F=F_{1}} {\displaystyle F=F_{1}}. As a consequence that degree is additive with respect to multiplication of functions and only function of degree d F < 1 {\displaystyle d_{F}<1} {\displaystyle d_{F}<1} is F = 1 {\displaystyle F=1} {\displaystyle F=1}, every function F ≠ 1 can be written as a product of primitive functions. However, uniqueness of this factorization is still unproven.

The prototypical example of an element in S is the Riemann zeta function.[3] 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 , Δ ) = ∑ n = 1 ∞ τ ( n ) / n 11 / 2 n s , {\displaystyle L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {\tau (n)/n^{11/2}}{n^{s}}},} {\displaystyle L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {\tau (n)/n^{11/2}}{n^{s}}},}

where τ ( n ) {\textstyle \tau (n)} {\textstyle \tau (n)} is the Ramanujan tau function.[4] This example can be considered a "normalized" or "shifted" _L_-function for the original Ramanujan L-function, defined as

L ( s ) = ∑ n = 1 ∞ τ ( n ) n s , {\displaystyle L(s)=\sum _{n=1}^{\infty }{\frac {\tau (n)}{n^{s}}},} {\displaystyle L(s)=\sum _{n=1}^{\infty }{\frac {\tau (n)}{n^{s}}},}

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

( 1 2 π ) s Γ ( s ) L ( s ) = ( 1 2 π ) 12 − s Γ ( 12 − s ) L ( 12 − s ) {\displaystyle ({\frac {1}{2\pi }})^{s}\Gamma (s)L(s)=({\frac {1}{2\pi }})^{12-s}\Gamma (12-s)L(12-s)} {\displaystyle ({\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 zeros on the line R e ( s ) = 6 {\textstyle Re(s)=6} {\textstyle Re(s)=6}.

All known examples are automorphic _L_-functions, and the reciprocals of Fp(s) are polynomials in p_−_s of bounded degree.[5]

Selberg's conjectures

[edit]

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

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

It is conjectured that Selberg class is equal to class of automorphic _L_-functions. Primitive functions are expected to be associated with irreducible automorphic representations.

It is conjectured that all reciprocals of factors Fp(s) of the Euler products are polynomials in p_−_s of bounded degree.

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

Consequences of the conjectures

[edit]

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

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.[10][11][12]

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.

  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. ^ The zeros on the boundary are counted with half-multiplicity.
  3. ^ Murty 2008
  4. ^ Murty 2008
  5. ^ Murty 1994
  6. ^ Conrey & Ghosh 1993, § 4
  7. ^ A celebrated conjecture of Dedekind asserts that for any finite algebraic extension F {\displaystyle F} {\displaystyle F} of Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }, the zeta function ζ F ( s ) {\displaystyle \zeta _{F}(s)} {\displaystyle \zeta _{F}(s)} is divisible by the Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} {\displaystyle \zeta (s)}. That is, the quotient ζ F ( s ) / ζ ( s ) {\displaystyle \zeta _{F}(s)/\zeta (s)} {\displaystyle \zeta _{F}(s)/\zeta (s)} is entire. More generally, Dedekind conjectures that if K {\displaystyle K} {\displaystyle K} is a finite extension of F {\displaystyle F} {\displaystyle F}, then ζ K ( s ) / ζ F ( s ) {\displaystyle \zeta _{K}(s)/\zeta _{F}(s)} {\displaystyle \zeta _{K}(s)/\zeta _{F}(s)} should be entire. This conjecture is still open.
  8. ^ 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 representation as predicted by the Langlands conjectures.Murty 1994, Theorem 4.3
  9. ^ Murty 1994, Lemma 4.2
  10. ^ Aistleitner, Christoph; Pańkowski, Łukasz (2017). "Large values of L-functions from the Selberg class". J. Math. Anal. Appl. 446 (1): 345–364. arXiv:1507.06066. doi:10.1016/j.jmaa.2016.08.044.
  11. ^ Palojärvi, Neea; Simonič, Aleksander (2022). "Conditional estimates for L-functions in the Selberg class". arXiv:2211.01121 [math.NT].
  12. ^ Palojärvi, Neea; Simonič, Aleksander (2024). "Conditional upper and lower bounds for L-functions in the Selberg class close to the critical line". arXiv:2410.22711 [math.NT].