Value-Distribution of L-Functions
L-functions are important objects in modern number theory. They are gen- ating functions formed out of local data associated with either an arithmetic object or with an automorphic form. They can be attached to smooth p- jective varieties de?ned over number ?elds, to irreducible (complex orp-adic) representations of the Galois group of a number ?eld, to a cusp form or to an irreducible cuspidal automorphic representation. All theL-functions have in common that they can be described by an Euler product, i. e., a product takenoverprimenumbers. Inviewoftheuniqueprimefactorizationofintegers L-functions also have a Dirichlet series representation. The famous Riemann zeta-function ? ?1 1 1 ?(s)= = 1? s s n p n=1 pprime may be regarded as the prototype. L-functions encode in their val- distribution information on the underlying arithmetic or algebraic structure that is often not obtainable by elementary or algebraic methods. For instance, Dirichlet’s class number formula gives information on the deviation from unique prime factorization in the ring of integers of quadratic number ?elds by the values of certain DirichletL-functionsL(s,?) ats=1. In parti- lar, the distribution of zeros ofL-functions is of special interest with respect to many problems in multiplicative number theory. A ?rst example is the Riemann hypothesis on the non-vanishing of the Riemann zeta-function in the right half of the critical strip and its impact on the distribution of prime numbers. Another example areL-functionsL(s,E) attached to elliptic curves E de?ned over Q.
Offers recent results in the value-distribution theory of L-functionsUniversality is proved for polynomial Euler productsThe authors offer the conjecture that all reasonable L-functions are universal