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{{Short description|Axiomatic definition of a class of L-functions}}
[[File:Atle_Selberg.jpg|thumb|Atle Selberg]]
In [[mathematics]], the '''Selberg class''' is an [[axiom]]atic definition of a class of [[L-function|''L''-function]]s.  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 function]]s.  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 form]]s and the [[Riemann hypothesis]].  The class was defined by [[Atle Selberg]] in {{harv|Selberg|1992}}, who preferred not to use the word "axiom" that later authors have employed.<ref>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.</ref>

==Definition==
The formal definition of the class ''S'' is the set of all [[Dirichlet series]]

:<math>F(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}</math>

absolutely convergent for Re(''s'')&nbsp;&gt;&nbsp;1 that satisfy four axioms (or assumptions as Selberg calls them):

{{ordered list
|1=  [[analytic function|Analyticity]]: <math>F(s)</math> has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s equals 1.
|2=  [[Ramanujan conjecture]]: ''a''<sub>1</sub> = 1 and <math>a_n \ll_\varepsilon n^\varepsilon</math> for any ε&nbsp;&gt;&nbsp;0;

|3=  [[Functional equation (L-function)|Functional equation]]: there is a gamma factor of the form

:<math>\gamma(s)=Q^s\prod_{i=1}^k \Gamma (\omega_is+\mu_i)</math>
where ''Q'' is real and positive, Γ the [[gamma function]], the &omega;<sub>''i''</sub> real and positive, and the &mu;<sub>''i''</sub> complex with non-negative real part, as well as a so-called root number

:<math>\alpha\in\mathbb C,\;|\alpha|=1</math>,

such that the function

:<math>\Phi(s) = \gamma(s) F(s)\,</math>

satisfies

:<math>\Phi(s)=\alpha\,\overline{\Phi(1-\overline{s})};</math>

|4=  [[Euler product]]: For Re(''s'')&nbsp;&gt;&nbsp;1, ''F''(''s'') can be written as a product over primes:

:<math>F(s)=\prod_p F_p(s)</math>

with

:<math> F_p(s)=\exp\left(\sum_{n=1}^\infty\frac{b_{p^n}}{p^{ns}}\right)</math>

and, for some θ &lt; 1/2,

:<math>b_{p^n}=O(p^{n\theta}).\,</math>
}}

===Comments on definition===
The condition that the real part of &mu;<sub>''i''</sub> be non-negative is because there are known ''L''-functions that do not satisfy the [[Riemann hypothesis]] when &mu;<sub>''i''</sub> is negative. Specifically, there are [[Maass form]]s associated with exceptional eigenvalues, for which the [[Ramanujan–Petersson conjecture]] holds, and have a functional equation, but do not satisfy the Riemann hypothesis.

The condition that &theta; &lt; 1/2 is important, as the &theta; = 1 case includes <math>(1-2^{-s})(1-2^{1-s})</math> whose zeros are not on the critical line.

Without the condition <math>a_n \ll_\varepsilon n^\varepsilon</math> there would be <math>L(s+1/3,\chi_4)L(s-1/3,\chi_4)</math> which violates the Riemann hypothesis.

It is a consequence of 4. that the ''a<sub>n</sub>'' are [[multiplicative function|multiplicative]] and that
:<math>F_p(s)=\sum_{n=0}^\infty\frac{a_{p^n}}{p^{ns}}\text{ for Re}(s)>0.</math>

===Examples===
The prototypical example of an element in ''S'' is the [[Riemann zeta function]].<ref>{{harvnb|Murty|2008}}</ref> Dirichlet ''L''-functions associated with primitive characters modulo <math>q \geq 2</math> belong to the Selberg class, too. Another example, is the ''L''-function of the [[modular discriminant]] Δ 
:<math>L(s,\Delta)=\sum_{n=1}^\infty\frac{a_n}{n^s}</math>
where <math>a_n=\tau(n)/n^{11/2}</math> and τ(''n'') is the [[Ramanujan tau function]].<ref>{{harvnb|Murty|2008}}</ref>

All known examples are [[automorphic L-function|automorphic ''L''-function]]s, and the reciprocals of ''F<sub>p</sub>''(''s'') are polynomials in ''p''<sup>&minus;''s''</sup> of bounded degree.<ref>{{harvnb|Murty|1994}}</ref>

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.<ref>{{cite journal |title= On the structure of the Selberg class, VII. |author= Jerzy Kaczorowski |author2= Alberto Perelli |name-list-style= amp |journal= Annals of Mathematics |volume= 173 |year= 2011 |pages= 1397–1441 |url= http://annals.math.princeton.edu/wp-content/uploads/annals-v173-n3-p04-p.pdf | doi = 10.4007/annals.2011.173.3.4|doi-access= free }}</ref>

==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 {{nowrap|1 &minus; ''A'' ≤ Re(''s'') ≤ ''A''}}. Denoting the number of non-trivial zeroes of ''F'' with {{nowrap|0 ≤ Im(''s'') ≤ ''T''}} by ''N<sub>F</sub>''(''T''),<ref>The zeroes on the boundary are counted with half-multiplicity.</ref> Selberg showed that
:<math>N_F(T)=d_F\frac{T\log(T+C)}{2\pi}+O(\log T).</math>
An explicit version of the result was proven by Palojärvi.<ref name="Pal2019">{{cite journal |last1=Palojärvi |first1=Neea |title=On the explicit upper and lower bounds for the number of zeros of the Selberg class |journal=J. Number Theory |date=2019 |volume=194 |pages=218–250 |doi=10.1016/j.jnt.2018.07.006|arxiv=1709.07170 }}</ref> Here, ''d<sub>F</sub>'' is called the '''degree''' (or '''dimension''') of ''F''. It is given by<ref>While the ω<sub>''i''</sub> are not uniquely defined by ''F'', Selberg's result shows that their sum is well-defined.</ref>
:<math>d_F=2\sum_{i=1}^k\omega_i.</math>
It can be shown that ''F''&nbsp;=&nbsp;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
:<math>d_{FG}=d_F+d_G.</math>
A function {{nowrap|''F'' ≠ 1}} in ''S'' is called '''primitive''' if whenever it is written as ''F''&nbsp;=&nbsp;''F''<sub>1</sub>''F''<sub>2</sub>, with ''F<sub>i</sub>'' in ''S'', then ''F''&nbsp;=&nbsp;''F''<sub>1</sub> or ''F''&nbsp;=&nbsp;''F''<sub>2</sub>. If ''d<sub>F</sub>''&nbsp;=&nbsp;1, then ''F'' is primitive. Every function {{nowrap|''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-function|Dirichlet ''L''-functions]] of primitive Dirichlet characters. Assuming conjectures 1 and 2 below, ''L''-functions of [[irreducible representation|irreducible]] [[cuspidal representation|cuspidal]] [[automorphic representation]]s that satisfy the Ramanujan conjecture are primitive.<ref>{{harvnb|Murty|1994|loc=Lemma 4.2}}</ref>

==Selberg's conjectures==

In {{harv|Selberg|1992}}, Selberg made conjectures concerning the functions in ''S'':
*Conjecture 1: For all ''F'' in ''S'', there is an integer ''n<sub>F</sub>'' such that <math display="block">\sum_{p\leq x}\frac{|a_p|^2}{p}=n_F\log\log x+O(1)</math>  and ''n<sub>F</sub>''&nbsp;=&nbsp;1 whenever ''F'' is primitive.
*Conjecture 2: For distinct primitive ''F'',&nbsp;''F''′&nbsp;∈&nbsp;''S'', <math display="block">\sum_{p\leq x} \frac{a_p\overline{a_p^\prime}}{p}=O(1).</math>
*Conjecture 3: If ''F'' is in ''S'' with primitive factorization <math display="block">F = \prod_{i=1}^m F_i,</math> χ is a primitive Dirichlet character, and the function <math display="block"> F^\chi(s) = \sum_{n=1}^\infty\frac{\chi(n)a_n}{n^s}</math> is also in ''S'', then the functions ''F<sub>i</sub>''<sup>χ</sup> are primitive elements of ''S'' (and consequently, they form the primitive factorization of ''F''<sup>χ</sup>).
*Generalized Riemann hypothesis for ''S'': For all ''F'' in ''S'', the non-trivial zeroes of ''F'' all lie on the line Re(''s'')&nbsp;=&nbsp;1/2.

===Consequences of the conjectures===

Conjectures 1 and 2 imply that if ''F'' has a pole of order ''m'' at ''s''&nbsp;=&nbsp;1, then ''F''(''s'')/ζ(''s'')<sup>''m''</sup> is entire. In particular, they imply Dedekind's conjecture.<ref>A celebrated conjecture of Dedekind asserts that for any finite algebraic extension
<math>F</math> of <math>\mathbb{Q}</math>, the zeta function <math>\zeta_F (s)</math> is divisible by the Riemann zeta function <math>\zeta (s)</math>. That is, the quotient <math>\zeta_F (s)/\zeta(s)</math> is entire. More generally, Dedekind conjectures that if <math>K</math> is a finite extension of <math>F</math>, then <math>\zeta_K (s)/\zeta_F(s)</math> should be entire. This conjecture is still open.</ref>

[[M. Ram Murty]] showed in {{harv|Murty|1994}} that conjectures 1 and 2 imply the [[Artin conjecture (L-functions)|Artin conjecture]]. In fact, Murty showed that [[Artin L-function|Artin ''L''-functions]] corresponding to irreducible representations of the [[Galois group]] of a [[solvable extension]] of the rationals are [[automorphic representation|automorphic]] as predicted by the [[Langlands conjectures]].<ref>{{harvnb|Murty|1994|loc=Theorem 4.3}}</ref>

Combined with the Generalized Riemann hypothesis, different versions of Conjectures 1 and 2 imply certain growth rates for the function and its logarithmic derivative.<ref name="AinPan2017">{{cite journal |last1=Aistleitner |first1=Christoph |last2=Pańkowski |first2=Łukasz |title=Large values of L-functions from the Selberg class |journal=J. Math. Anal. Appl. |date=2017 |volume=446 |issue=1 |pages=345–364|doi=10.1016/j.jmaa.2016.08.044 |arxiv=1507.06066 }}</ref><ref name="PalSim2022">{{cite arXiv |last1=Palojärvi |first1=Neea |last2=Simonič |first2=Aleksander |title=Conditional estimates for L-functions in the Selberg class |date=2022 |class=math.NT |eprint=2211.01121 }}</ref><ref name="PalSim2024">{{cite arXiv |last1=Palojärvi |first1=Neea |last2=Simonič |first2=Aleksander |title=Conditional upper and lower bounds for L-functions in the Selberg class close to the critical line |date=2024 |class=math.NT |eprint=2410.22711 }}</ref>

The functions in ''S'' also satisfy an analogue of the [[prime number theorem]]: ''F''(''s'') has no zeroes on the line Re(''s'')&nbsp;=&nbsp;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<sub>F</sub>''&nbsp;=&nbsp;1.<ref>{{harvnb|Conrey|Ghosh|1993|loc=§ 4}}</ref>

==See also==
* [[List of zeta functions]]

==Notes==

{{reflist}}

==References==
* {{Citation | last=Selberg | first=Atle | title=Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989) | publisher=Univ. Salerno | location=Salerno | mr=1220477 | zbl=0787.11037 | year=1992 | chapter=Old and new conjectures and results about a class of Dirichlet series | pages=367–385}} Reprinted in Collected Papers, vol '''2''', Springer-Verlag, Berlin (1991)
*{{Citation
| last1=Conrey
| first1=J. Brian
| author-link=Brian Conrey
| last2=Ghosh
| first2=Amit
| title=On the Selberg class of Dirichlet series: small degrees
| year=1993
| journal=Duke Mathematical Journal
| volume=72
| number=3
| pages=673–693
| arxiv=math.NT/9204217
| mr=1253620 | zbl=0796.11037
 | doi=10.1215/s0012-7094-93-07225-0
}}

*{{Citation
| last=Murty
| first=M. Ram
| author-link=M. Ram Murty
| title=Selberg's conjectures and Artin ''L''-functions
| year=1994
| journal=Bulletin of the American Mathematical Society |series=New Series
| volume=31
| number=1
| pages=1–14
| mr=1242382 | arxiv=math/9407219 | zbl=0805.11062
 | doi=10.1090/s0273-0979-1994-00479-3
| s2cid=265909
}}

*{{Citation
| last=Murty
| first=M. Ram
| author-link=M. Ram Murty
| title=Problems in analytic number theory
| year=2008
| edition=Second
| publisher=[[Springer-Verlag]]
| series=[[Graduate Texts in Mathematics]], Readings in Mathematics
| volume=206
| mr=2376618 | zbl=1190.11001 
| isbn=978-0-387-72349-5
| doi=10.1007/978-0-387-72350-1 | at=Chapter 8
}}

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