Axiomatic Theory of L-Functions: the Selberg Class

Kaczorowski, Jerzy

doi:10.1007/978-3-540-36364-4_4

Cited by 73 publications

(81 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…A convenient framework for such an analysis is provided by the theory of the Selberg class S. We refer to [2][3][4][5] for the basic definitions and mention only that S consists of the Dirichlet series …”

Section: ϕ(N F) = C(f)x 2 + O(x(log(2x)) D )mentioning

confidence: 99%

On a generalization of the Euler totient function

Kaczorowski

2012

Monatsh Math

Self Cite

View full text Add to dashboard Cite

For a general polynomial Euler product F(s) we define the associated Euler totient function ϕ(n, F) and study its asymptotic properties. We prove that for F(s) belonging to certain subclass of the Selberg class of L-functions, the error term in the asymptotic formula for the sum of ϕ(n, F) over positive integers n ≤ x behaves typically as a linear function of x. We show also that for the Riemann zeta function the square mean value of the error term in question is minimal among all polynomial Euler products from the Selberg class, and that this property uniquely characterizes ζ(s).

show abstract

Section: ϕ(N F) = C(f)x 2 + O(x(log(2x)) D )mentioning

confidence: 99%

On a generalization of the Euler totient function

Kaczorowski

2012

Monatsh Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…There are currently several proofs of Hamburger's theorem, based on few principles. Some proofs are based on an analogous uniqueness statement for certain theta functions, other proofs exploit certain periodicity properties induced by the special form of (16). We refer to Chapter II of Titchmarsh [45] and to Piatetski-Shapiro and Raghunathan [41] for a discussion of Hamburger's theorem, and to Section 3 for a more recent approach in a general framework, giving Theorem 2.1 as a very special case.…”

Section: Classical Converse Theoremsmentioning

confidence: 99%

“…Since L-functions come usually from finite dimensional structures, we see that converse theorems must involve non-trivial information. Moreover, the dimension of W is highly sensitive to variations in the data of the functional equation; for example, slight modifications to (16) give already a different output. Indeed, write F (s) = F (s) and consider the space W of solutions of the functional equation…”

Section: Classical Converse Theoremsmentioning

confidence: 99%

“…We refer to the original paper by Selberg [43], to the early papers on the subject by Conrey-Ghosh [10] and Ram Murty [34] and to the surveys of Kaczorowski-Perelli [20], Kaczorowski [16] and Perelli [38], [39], [40] for the basic theory of the Selberg class. The reader may find there all the results reported below, if no reference is given.…”

Section: Converse Theorems For the Selberg Classmentioning

confidence: 99%

See 1 more Smart Citation

Converse theorems: from the Riemann zeta function to the Selberg class

Perelli

2016

Boll Unione Mat Ital

View full text Add to dashboard Cite

Abstract. This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions.Mathematics Subject Classification (2000): 11F66, 11M41

show abstract

“…In [6] and [7] we defined and studied the invariants of the Selberg class S (to be precise, of the extended Selberg class S ). We refer to our survey papers [3], [5], [9] and [10] for the definitions and basic properties of the classes S and S . Here we recall that S is the class of non-identically vanishing Dirichlet series Γ (λ j s + µ j )F (s) = γ(s)F (s), say, with r ≥ 0, Q > 0, λ j > 0 and µ j ≥ 0 (r = 0 means that there are no Γ -factors).…”

mentioning

confidence: 99%