scholarly journals A Generalization of the Secant Zeta Function as a Lambert Series

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
H.-Y. Li ◽  
B. Maji ◽  
T. Kuzumaki
Keyword(s):  
Zeta Function ◽  
Lambert Series ◽  
Dedekind Eta Function ◽  
Irrational Numbers ◽  
Quadratic Irrational ◽  
Modular Transformation

Recently, Lalín, Rodrigue, and Rogers have studied the secant zeta function and its convergence. They found many interesting values of the secant zeta function at some particular quadratic irrational numbers. They also gave modular transformation properties of the secant zeta function. In this paper, we generalized secant zeta function as a Lambert series and proved a result for the Lambert series, from which the main result of Lalín et al. follows as a corollary, using the theory of generalized Dedekind eta-function, developed by Lewittes, Berndt, and Arakawa.

Dedekind sums and Hecke operators

1980 ◽  
Vol 88 (1) ◽  
pp. 11-14 ◽  
Author(s):  
L. Alayne Parson
Keyword(s):  
Elementary Proof ◽  
Hecke Operators ◽  
Transformation Formula ◽  
Dedekind Sums ◽  
Lambert Series ◽  
Dedekind Eta Function ◽  
Dedekind Sum ◽  
Modular Transformation ◽  
Special Case

By considering the action of the Hecke operators on the logarithm of the Dedekind eta function together with the modular transformation formula for this function, Knopp (8) proved an extension of an identity of Dedekind for the classical Dedekind sums first mentioned by H. Petersson. By looking at the action of the Hecke operators on certain Lambert series studied by Apostol(l) together with the transformation formulae for these series, Parson and Rosen (9) established an analogous identity for a type of generalized Dedekind sum. A special case of this identity was initially proved by Carlitz(6). In this note an elementary proof of these identities is given. The Hecke operators are applied directly to the Dedekind sums without invoking the transformation formulae for the logarithm of the eta function or for the Lambert series. (Recently, L. Goldberg has given another elementary proof of Knopp's identity.)

Properties of real quadratic irrational numbers under the action of the group H = áx, y: x2 = y4 = 1ñ

2005 ◽  
Vol 42 (4) ◽  
pp. 371-386
Author(s):  
M. Aslam Malik ◽  
S. M. Husnine ◽  
Abdul Majeed
Keyword(s):  
Quadratic Field ◽  
Infinite Group ◽  
Algebraic Structures ◽  
Real Quadratic Field ◽  
Irrational Numbers ◽  
The Real ◽  
Basic Properties ◽  
Quadratic Irrational ◽  
Real Quadratic

Studying groups through their actions on different sets and algebraic structures has become a useful technique to know about the structure of the groups. The main object of this work is to examine the action of the infinite group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H = \langle x,y : x^{2} = y^{4} = 1\rangle$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $x (z) = \frac{-1}{2z}$ \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $y (z) = \frac{-1}{2(z+1)}$ \end{document} on the real quadratic field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} and find invariant subsets of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $H$ \end{document}. We also discuss some basic properties of elements of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathbb{Q}\left(\sqrt{n}\,\right)$ \end{document} under the action of the group H.

Coset Diagram for the Action of Picard Group on $\mathbb{Q}(i,\sqrt{3})$

2016 ◽  
Vol 23 (01) ◽  
pp. 33-44 ◽  
Author(s):  
Qaiser Mushtaq ◽  
Saima Anis
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Picard Group ◽  
Closed Path ◽  
Irrational Numbers ◽  
Unique Pattern ◽  
Graphical Interpretation ◽  
Quadratic Irrational ◽  
Coset Diagrams ◽  
Real Quadratic

In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group Γ=PSL(2,ℤ[i]) on [Formula: see text]. Graphical interpretation of amalgamation of the components of Γ is also given. Some elements [Formula: see text] of [Formula: see text] and their conjugates [Formula: see text] over ℚ(i) have different signs in the orbits of the biquadratic field [Formula: see text] when acted upon by Γ. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Γα, and they form a closed path which is the only closed path in the orbit Γα. We also devise a procedure to obtain ambiguous numbers of the form [Formula: see text], where b is a positive integer.

Modular group acting on real quadratic fields

1988 ◽  
Vol 37 (2) ◽  
pp. 303-309 ◽  
Author(s):  
Q. Mushtaq
Keyword(s):  
Positive Integer ◽  
Finite Number ◽  
Modular Group ◽  
Quadratic Fields ◽  
Closed Path ◽  
Irrational Numbers ◽  
Real Quadratic Fields ◽  
Quadratic Irrational ◽  
Coset Diagrams ◽  
Real Quadratic

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.

scholarly journals Generalized Lambert series and arithmetic nature of odd zeta values

2019 ◽  
Vol 150 (2) ◽  
pp. 741-769 ◽  
Author(s):  
Atul Dixit ◽  
Bibekananda Maji
Keyword(s):  
Lambert Series ◽  
Dedekind Eta Function ◽  
Euler’S Constant ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook ◽  
Zeta Values ◽  
Euler's Constant

AbstractIt is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.

CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)

2009 ◽  
Vol 05 (05) ◽  
pp. 765-778 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Eisenstein Series ◽  
Congruence Subgroup ◽  
Gauss Sums ◽  
Triple Product ◽  
Simple Construction ◽  
Dedekind Eta Function ◽  
Modular Transformation ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Extra Difficulty

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

scholarly journals On some Lambert-like series

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
P Agarwal ◽  
S Kanemitsu ◽  
T Kuzumaki
Keyword(s):  
Power Series ◽  
Zeta Function ◽  
Rational Function ◽  
Laurent Series ◽  
Zeta Functions ◽  
Limit Function ◽  
Lambert Series ◽  
Radial Limits ◽  
Lerch Zeta Function ◽  
International Audience

International audience In this note, we study radial limits of power and Laurent series which are related to the Lerch zeta-function or polylogarithm function. As has been pointed out in [CKK18], there have appeared many instances in which the imaginary part of the Lerch zeta-function was considered by eliminating the real part by considering the odd part only. Mordell studied the properties of the power series resembling Lambert series, and in particular considered whether the limit function is a rational function or not. Our main result is the elucidation of the threshold case of b_n = 1/n studied by Mordell [Mor63], revealing that his result is the odd part of Theorem 1.1 in view of the identities (1.9), (1.5). We also refer to Lambert series considered by Titchmarsh [Tit38] in connection with Estermann's zeta-functions.

On the Convergence of a Class of Nearly Alternating Series

2007 ◽  
Vol 59 (1) ◽  
pp. 85-108
Author(s):  
J. H. Foster ◽  
Monika Serbinowska
Keyword(s):  
Continued Fraction ◽  
Partial Sums ◽  
Real Numbers ◽  
Irrational Numbers ◽  
Quadratic Irrational

AbstractLet C be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If α is of the first type and (ck) ∈ C, then ∑(—1)⎿ck⏌ converges if and only if ck log k → 0. If α is of the second type and (ck) ∈ C, then ∑(—1)⎿ck⏌ converges if and only if ∑ ck/k converges. An example of a quadratic irrational of the first type is and an example of the second type is . The analysis of this problem relies heavily on the representation of α as a simple continued fraction and on properties of the sequences of partial sums and double partial sums .

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