Zeta function universality

2013-05-01T20:12:34Z

87.91.111.12: /* Discussion */

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}}.

==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'') > 1 that satisfy four axioms:

<ol>
<li> [[analytic function|Analyticity]]: the function (''s'' − 1)''m''''F''(''s'') is an [[entire function]] of finite [[order of an entire function|order]] for some non-negative integer ''m'';

<li> [[Ramanujan conjecture]]: ''a''1 = 1 and <math>a_n \ll_\epsilon n^\epsilon</math> for any ε > 0;

<li> [[Functional equation (L-function)|Functional equation]]: there is a gamma factor of the form

:<math>\gamma(s)=e^{i\phi}Q^s
\prod_{i=1}^k \Gamma (\omega_is+\mu_i)</math>
where φ is real, ''Q'' real and positive, Γ is the [[gamma function]], the ω1 real and positive, and the μ''i'' complex with non-negative real part, so that the function

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

satisfies

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

<li> [[Euler product]]: ''F''(''s'') can be written as a product over primes:

:<math>F(s)=\prod_p F_p(s)\text{ for Re}(s)>1\,</math>

with

:<math>\log F_p(s)=\sum_{n=0}^\infty \frac{b_{p^n}}{p^{ns}}</math>

and, for some θ < 1/2,

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

===Comments on definition===

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 cusp form]]s 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|Dirichlet eta-function]], which violates the Riemann hypothesis.<ref>{{harvnb|Conrey|Ghosh|1993|loc=§1}}</ref>

It is a consequence of 4. that the ''an'' 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>{{cite book | title=In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes | first=Michel Laurent | last=Lapidus | publisher=[[American Mathematical Society]] | year=2008 | isbn=0821842226 | zbl=1150.11003 | page=389 }}</ref> 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> Additionally, if ''F'' is in ''S'' and χ is a [[primitive Dirichlet character]], then ''F''χ defined by
:<math>F^\chi(s)=\sum_{n=1}^\infty\frac{\chi(n)a_n}{n^s}</math>
is also in ''S''.

All known examples are [[automorphic L-function|automorphic ''L''-function]]s, and the reciprocals of ''Fp''(''s'') are polynomials in ''p''−''s'' of bounded degree.<ref>{{harvnb|Murty|1994}}</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 − ''A'' ≤ Re(''s'') ≤ ''A''}}. Denoting the number of non-trivial zeroes of ''F'' with {{nowrap|0 ≤ Im(''s'') ≤ ''T''}} by ''NF''(''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>
Here, ''dF'' is called the '''degree''' (or '''dimension''') of ''F''. It is given by<ref>While the ω''i'' 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'' = 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'' = ''F''1''F''2, with ''Fi'' in ''S'', then ''F'' = ''F''1 or ''F'' = ''F''2. If ''dF'' = 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 ''nF'' such that
::<math>\sum_{p\leq x}\frac{|a_p|^2}{p}=n_F\log\log x+O(1)</math>
:and ''nF'' = 1 whenever ''F'' is primitive.
*Conjecture 2: For distinct primitive ''F'', ''F''′ ∈ ''S'',
::<math>\sum_{p\leq x}\frac{a_pa_p^\prime}{p}=O(1).</math>
*Conjecture 3: If
::<math>F=\prod_{i=1}^mF_i</math>
:is a factorization of ''F'' into primitive functions and χ is a primitive Dirichlet character, then
::<math>F^\chi=\prod_{i=1}^mF_i^\chi</math>
:and the ''Fi''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'')''m'' is entire. In particular, they imply [[Dedekind's conjecture]].

[[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>

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.<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
| last=Conrey
| first=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
}}

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

*{{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
}}
*{{citation | last=Ivić | first=Aleksandar | title=The theory of Hardy's ''Z''-function | series=Cambridge Tracts in Mathematics | volume=196 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2013 | isbn=978-1-107-02883-8 | zbl=pre06093527 }}

{{L-functions-footer}}

[[Category:Zeta and L-functions]]

formulasearchengine - User contributions [en]

Zeta function universality