ON THE MODULUS OF THE SELBERG ZETA-FUNCTIONS IN THE CRITICAL STRIP

We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functions ZPSL(2,Z)(s) and ZC (s) in the critical strip 0 < σ < 1. The functions ZPSL(2,Z)(s) and ZC (s) are defined on the modular group and on the compact Riemann surface, respectively.

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PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

International Journal of Number Theory ◽

10.1142/s1793042109002419 ◽

2009 ◽

Vol 05 (05) ◽

pp. 845-857 ◽

Cited By ~ 3

Author(s):

MARVIN KNOPP ◽

GEOFFREY MASON

Keyword(s):

Riemann Surface ◽

Modular Form ◽

Finite Index ◽

Modular Forms ◽

Modular Group ◽

Compact Riemann Surface ◽

Eichler Cohomology

We make a detailed study of the generalized modular forms of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup Γ of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to Γ is the divisor of a modular form (with unitary character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.

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ON LOGARITHMIC DERIVATIVES OF ZETA FUNCTIONS IN FAMILIES OF GLOBAL FIELDS

International Journal of Number Theory ◽

10.1142/s1793042111005015 ◽

2011 ◽

Vol 07 (08) ◽

pp. 2139-2156 ◽

Cited By ~ 1

Author(s):

PHILIPPE LEBACQUE ◽

ALEXEY ZYKIN

Keyword(s):

Error Term ◽

Function Fields ◽

Zeta Functions ◽

Number Fields ◽

Global Fields ◽

And Function ◽

Derivatives Of ◽

Logarithmic Derivatives ◽

Siegel Theorem

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

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On logarithmic derivatives of zeta functions in families of global fields

Doklady Mathematics ◽

10.1134/s1064562410020109 ◽

2010 ◽

Vol 81 (2) ◽

pp. 201-203

Author(s):

A. I. Zykin ◽

P. Lebacque

Keyword(s):

Zeta Functions ◽

Global Fields ◽

Derivatives Of ◽

Logarithmic Derivatives

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Weierstrass Points at the Cusps of Γ0(16p) and Hyperellipticity of Γ0(n)

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-103-6 ◽

1971 ◽

Vol 23 (6) ◽

pp. 960-968 ◽

Cited By ~ 2

Author(s):

H. Larcher

Keyword(s):

Riemann Surface ◽

Positive Integer ◽

Modular Group ◽

Compact Riemann Surface ◽

Weierstrass Point ◽

Weierstrass Points ◽

Linear Fractional Transformations ◽

Fixed Positive Integer

For a fixed positive integer n we consider the subgroup Γ0(n) of the modular group Γ(l). Γ0(n) consists of all linear fractional transformations L: z → (az + b)/(cz + d) with rational integers a, b, c, d, determinant ad – bc = 1, and c ≡ 0(mod n). If ℋ = {z|z = x + iy, x and y real and y > 0} is the upper half of the z-plane then S0 = S0(n) = ℋ/Γ0(n), properly compactified, is a compact Riemann surface whose genus we denote by g(n). A point P of a Riemann surface S of genus g is called a Weierstrass point if there exists a function on S that has a pole of order α ≦ g at P and is regular everywhere else on S.Lehner and Newman started the search for Weierstrass points of S0 (or, loosely, of Γ0(n)).

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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