On the Selberg class of Dirichlet series: small degrees

Abstract

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of Riemann hypothesis will hold, (ii) that if in addition the function has a simple pole at the point s=1, then it must be a product of the Riemann zeta-function and another Dirichlet series with similar properties, and (iii) that a type of converse theorem holds, namely that all such Dirichlet series can be obtained by considering Mellin transforms of automorphic forms associated with arithmetic groups.

Cited by in corpus (19)

• L-functions as distributions
• Converse theorems: from the Riemann zeta function to the Selberg class
• Twists and resonance of L-functions, II
• On the Northcott property for special values of L-functions
• A uniqueness property of general Dirichlet series
• Gaps between zeros of GL(2) $L$-functions
• On the Vertical Distribution of Values of $L$-functions in the Selberg Class
• Beyond the extended Selberg class: $d_F\le 1$
• Integral moments of L-functions
• Joint value distribution of $L$-functions on the critical line
• Hamiltonians arising from L-functions in the Selberg class
• Bernoulli Operators and Dirichlet Series
• Beurling Zeta Functions, Generalised Primes, and Fractal Membranes
• Local Behavior of Arithmetical Functions with Applications to Automorphic L-Functions
• Degree 1 elements of the Selberg class
• A relation between the zeros of different two $L$-functions which have the Euler product and functional equation
• Forbidden conductors and sequences of $\pm 1$s
• Riemann-Type Functional Equations -- Julia Line and Counting Formulae --
• A note on the invariants of the $L$-functions