The modular group and a hyperbolic beta integral

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Fermion masses and mixing from the double cover and metaplectic cover of the A5 modular group

Physical Review D ◽

10.1103/physrevd.103.095013 ◽

2021 ◽

Vol 103 (9) ◽

Author(s):

Chang-Yuan Yao ◽

Xiang-Gan Liu ◽

Gui-Jun Ding

Keyword(s):

Modular Group ◽

Double Cover ◽

Fermion Masses ◽

Metaplectic Cover

Identities on poly-Dedekind sums

Advances in Difference Equations ◽

10.1186/s13662-020-03024-x ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Hyunseok Lee ◽

Lee-Chae Jang

Keyword(s):

Modular Group ◽

Reciprocity Relation ◽

Dedekind Sums ◽

Transformation Behavior ◽

Dedekind Eta Function ◽

Bernoulli Function ◽

Bernoulli Functions

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽

10.3390/math9111254 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1254

Author(s):

Xue Han ◽

Xiaofei Yan ◽

Deyu Zhang

Keyword(s):

Asymptotic Formula ◽

Fourier Coefficient ◽

Cusp Form ◽

Riemann Hypothesis ◽

Modular Group ◽

Fourier Coefficients ◽

Mean Value ◽

Symmetric Square ◽

The Mean ◽

Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

The Picard group and the modular group

Proceedings of Groups – St Andrews 1985 ◽

10.1017/cbo9780511600647.014 ◽

2012 ◽

pp. 150-163

Author(s):

B. Fine

Keyword(s):

Modular Group ◽

Picard Group

An extension of the q-beta integral with applications

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2009.11.041 ◽

2010 ◽

Vol 365 (2) ◽

pp. 653-658 ◽

Cited By ~ 6

Author(s):

Mingjin Wang

Keyword(s):

Beta Integral

Some Remarks on q-Beta Integral

Proceedings of the American Mathematical Society ◽

10.2307/2043846 ◽

1982 ◽

Vol 85 (3) ◽

pp. 360 ◽

Cited By ~ 1

Author(s):

W. A. Al-Salam ◽

A. Verma

Keyword(s):

Beta Integral

Multiplier systems for the modular group on the 27-dimensional exceptional domain

Communications in Algebra ◽

10.1080/00927879808826210 ◽

1998 ◽

Vol 26 (5) ◽

pp. 1409-1417 ◽

Cited By ~ 4

Author(s):

Aloys Krieg ◽

Sebastian Walcher

Keyword(s):

Modular Group

PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150001806x ◽

1987 ◽

Vol 30 (1) ◽

pp. 143-151 ◽

Cited By ~ 8

Author(s):

David Singerman

Keyword(s):

Free Product ◽

Cyclic Group ◽

Modular Group ◽

Prime Power ◽

Group Actions ◽

Index 2 ◽

Image Position

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentationthe subgroup PSL(2, ℤ) being generated by UV and VW.

Lattice Subgroups of Normal Subgroups of Genus Zero of the Modular Group

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-24.3.449 ◽

1972 ◽

Vol s3-24 (3) ◽

pp. 449-469 ◽

Cited By ~ 4

Author(s):

A. W. Mason

Keyword(s):

Modular Group ◽

Normal Subgroups ◽

Genus Zero ◽

Lattice Subgroups