A CONCISE SURVEY OF THE SELBERG CLASS OF L-FUNCTIONS LI ZHENG ABSTRACT. In this survey paper, I first present some classical L-functions and its basic properties. Then I give the introduction of Selberg class of L-functions, and present some basic properties, important conjec- tures and consequences, and the relation with prime number theorem. Ever since Riemann’s revolutionary paper [1], the Riemann zeta function and its various generaliza- tions have been extensively studied by mathematicians for over a century. These functions are generally referred to as L-functions. Deep connections have been established between the properties of the L- functions and other theories (for example, prime number theory). Later in 1992, in attempt to capture the core properties of classical L-functions, Selberg gave an axiomatic characterization of what would be called general L-functions. This is paper is a concise survey for Selberg class of L-functions. 1. CLASSICAL L-FUNCTIONS In this section we will recall some common properties shared by a lot of classical L-functions. Proofs and details will be avoided; references will be provided. Also, we take the convention to write the variable s as σ Å i t. Example 1. Talking about L-functions, the first one to come to mind is of course Riemann’s ³ function, σ È which is defined, for 1, X Y ¡ ¡ ¡ ³(s) Æ n s Æ (1 ¡ p s) 1. n¸1 p It has a meromorphic continuation to the complex plane C, having a unique pole at s Æ 1. Setting ¡ ©(s) Æ ¼ s/2¡(s/2)³(s), we have the functional equation ©(s) Æ ©(1 ¡ s). For the theory of Riemann ³ function, see e.g. [2], [3]. Example 2. The most basic generalization of ³ function is Dirichlet L-function L(s,Â), which is defined by X Y ¡ ¡ ¡ L(s,Â) Æ Â(n)n s Æ (1 ¡ Â(p)p s) 1, for σ È 1, n¸1 p where  is a Dirichlet character modulo q, say. It has a meromorphic continuation to C with only a possible pole at s Æ 1. (This occurs precisely when  is principal.) It also satisfies a function equation under the assumption that  is primitive: Setting ³ ´ ³ ´ ¼ ¡(sÅa)/2 s Å a ¤(s,Â) Æ ¡ L(s,Â), k 2 Æ ¡  ¡ where a (1 ( 1))/2, then p i a k ¤(1 ¡ s,¯) Æ ¤(s,Â), ¿(Â) X p k ¼ where ¿(Â) Æ Â(n)ee in/k is the Gauss sum. (Notice that j¿(Â)j Æ k.) nÆ1 For detailed discussion, see e.g. [4], [20]. 1 Example 3. Dedekind ³ function. Let K be a number field of degree n Æ r1 Å2r2, where r1 is the number of real embeddings K ,! R, and r2 is the number of pairs of complex embeddings K ,! C. The Dedekind ³ function is defined by X Y ¡s ¡s ¡1 ³K (s) Æ N(I) Æ (1 ¡ N(p) ) , for σ È 1, I p where, in the sum, I runs over all non-zero ideals of K (by which we really mean the ideals of OK ); in the product, p runs over all non-zero prime ideals, and N Æ NK /Q is the norm. ³K has a meromorphic continuation to C, with a unique pole at s Æ 1. If we set ³ ´ jd j s K r1 r2 »K (s) Æ ¡ (s/2)¡ (s)³K (s), 4r2 ¼n where dK is the discriminant of K , then »K (s) Æ »K (1 ¡ s). See e.g. Ch. VII of Neukirch [13], Ch. 10 of Cohen [7]. Example 4. Hecke L-function. Let K be a number field and  a Hecke character. Then Hecke defined an L-function X Y ¡s ¡s ¡1 LK (s,Â) Æ Â(I)N(I) Æ (1 ¡ Â(p)N(p) ) , for σ È 1. I p This is a far reaching generalization of both Dirichlet L-function (as K Æ Q) and Dedekind ³ function (as  is the trivial character). It has a meromorphic continuation to C, with only a possible pole at s Æ 1, which occurs precisely when  is principle. Multiplying LK (s,Â) by a complicated gamma factor, one can achieve a functional equation. For details, see Ch. VII of [13]. Example 5. Artin L-function. Let K /k be a Galois extension of number fields and let (½,V ) be a repre- sentation of the Galois Group G Æ G(K /k). For each prime ideal p of k, pick a prime ideal P of K over p. Let DP Æ {t 2 G j t(P) Æ P} be the decomposition group of P. By passage to the quotient, there is a natural homomorphism DP ! G(K /k), where K Æ K /P, k Æ k/p. This homomorphism is surjective. The I kernal IP is called the inertial group of P. Then by passage to the quotient, DP/IP acts on V P , the fixed » subspace of IP. Since DP/IP Æ G(K /k), and K /k is an extension of finite fields, there is a natural notion of Frobenius element s(P/p) in DP/IP, which is the inverse image of the Frobenius element of G. Then we can define the Euler factor at p to be ¡1 ¡s IP Lp(s,½;K /k) Æ det (I ¡ N(p) ½jV (s(P/p))). Notice that this definition is independent of the choice of P because choosing a different P over p only changes s(P/p) to a conjugate element, thus does not change the determinant. Artin L-function is de- fined to be the product of Lp(s,½;K /k) as p runs over non-zero prime ideals of k. For properties of Artin L-function, see Ch. VII of [13], M.R. Murty, V.K. Murty [14]. Example 6. L-function associated to a modular form. The group SL2(Z) is called· the modular¸ group; the a b Hecke group ¡ (N) of level N is the subgroup of SL (Z) consisting all matrices with N j c. SL (Z) 0 2 c d 2 acts on the upper half-plane H Æ {z j Imz ¸ 0} by Möbius transformation: · ¸ a b az Å b : z 7! . c d cz Å d · ¸ a b For γ Æ 2 SL (Z), define j(γ,z) Æ cz Å d. Let k ¸ 0 be an integer. Define an operator [γ] on the c d 2 k space of meromorphic functions on H by ¡k (f [γ]k )(z) Æ j(γ,z) f (γ(z)). 2 ¼ The function q Æ e2 i z transforms H to the unit disk devoid of the origin. We introduce the infinity point 1 which corresponds to 0 via the above transformation. If f is holomorphic H, we can expand it at the infinity: X1 n f (z) Æ an q , called the q-expansion. nÆ¡1 A holomorphic function f : H ! C is called a modular form of weight k and level N if (i) f is invariant under the operation [γ]k for all γ 2 ¡0(N); (ii) f [®]k is holomorphic at 1 for all ® 2 SL2(Z). The second condition says that the coefficients an, n Ç 0, of f [®]k are all zero. If in addition f [®]k vanishes at 1 for all ® 2 SL2(Z), then f called a cusp form of weight k and level N. Let f be a modular form of weight k ¸ 1. Let X ¼ f (z) Æ a(n)qn, q Æ e2 i z , n¸0 be the q-expansion of f at the infinity. Then one can define an L-function X ¡ L(f ,s) Æ a(n)n s. n¸1 It can be extended to a meromorphic function on C, which is entire if f is a cusp form, or has a pole at s Æ k otherwise. For details, see Iwaniec, Kowalski [16]. We mention that there are also L-functions associated to general automorphic forms. (Loc. cit.) Example 7. L-function associated to elliptic curves. Let E/Q be an elliptic curve, with conductor N. Then E has stable reduction at all primes p away from divisors of N. It has a semistable reduction at primes p with p ∥ N, and unstable reduction at primes p with p2 j N. The local zeta function of E is given by 8 ¡ ¡ <>(1 ¡ a(p)p s Å p1 2s), if p - N; ¡ L (s,E) Æ (1 ¡ a(p)p s), if p ∥ N; p :> 1, if p2 j N, where a(p) Æ p Å 1 in the case where p - N, a(p) Ƨ1 when p ∥ N depending whether E has a split or non-split semistable reduction at p. Then the L-function associated to E is defined by Y L(s,E) Æ Lp (s,E). p See Silverman [17]. We mention that this is a special case of Hasse-Weil L-function, which is attached to an algebraic variety over a number field. 2. SELBERG CLASSOF L-FUNCTION In the first section, We have given several examples of what are classically called L-functions, which are of different nature: Examples 1, 2 are arthmetic; 3–5 are algebraic; 7 is geometric. It is natural to ask, what is an L-function? Are all L-functions already known? Of course, the answer to the second question depends on the answer to the first. Selberg, in attempt to study the properties of various L-functions in a unified way, introduced the Selberg class S in [5].