The Lerch zeta-function / Antanas Laurinčikas and Ramūnas Garunkštis.

Format:

Book

Author/Creator:

Laurinčikas, Antanas.

Contributor:

Garunkštis, Ramūnas.

Alumni and Friends Memorial Book Fund.

Language:

English

Subjects (All):

Functions, Zeta.

Euler products.

Physical Description:

viii, 189 pages ; 24 cm

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, [2002]

Contents:

1. Euler Gamma-Function 1

1.1 Definition and Analytic Continuation 1

1.2 Representation by an Infinite Product 3

1.3 Functional Equation 6

1.4 Complementary Formula 7

1.5 Asymptotic Formulas 9

1.6 Hypergeometric Function 12

2. Functional Equation 17

2.1 Definition of the Lerch Zeta-Function 17

2.2 Analytic Continuation 18

2.3 Functional Equation 21

2.4 Application of the Euler-Maclaurin Formula 23

3. Moments 31

3.1 Approximation of L ([lambda], [alpha], s) by a Finite Sum 31

3.2 Montgomery-Vaughan Theorem 35

3.3 Mean Square of L ([lambda], [alpha], s) 36

3.4 Mean Square of L ([lambda], [alpha], s) with Respect to [alpha] 39

4. Approximate Functional Equation 53

4.1 Proof of the Approximate Functional Equation 53

4.2 Application of the Approximate Functional Equation to the Mean Square of L ([lambda], [alpha], s) 59

5. Statistical Properties 71

5.1 Limit Theorems on the Complex Plane 71

5.2 Limit Theorems in the Space of Analytic Functions 80

5.3 Joint Limit Theorems in the Space of Analytic Functions 97

5.4 Limit Theorems in the Space of Analytic Functions with Rational [alpha] 102

6. Universality 111

6.1 Case of Transcendental [alpha] 111

6.2 Case of Rational [alpha] 117

6.3 Joint Universality of Lerch Zeta-Functions 121

6.4 Effectivization Problem of the Universality Theorem 130

7. Functional Independence 137

7.1 The One-Dimensional Case 137

7.2 Joint Functional Independence 140

8. Distribution of Zeros 143

8.1 Zero-Free Region on the Right 143

8.2 Zero-Free Regions on the Left 144

8.3 Number of Nontrivial Zeros 152

8.4 Estimates of the Number of Nontrivial Zeros 156

8.5 Sums over Nontrivial Zeros 170.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

1402010141

OCLC:

51093419