Expansion of the Inhomogeneous Modular Transformations with   Modular Group Properties

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.

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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

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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.

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Modular invariance in finite temperature Casimir effect

Journal of High Energy Physics ◽

10.1007/jhep10(2020)134 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Francesco Alessio ◽

Glenn Barnich

Keyword(s):

Partition Function ◽

Chemical Potential ◽

Casimir Effect ◽

Temperature Inversion ◽

Massless Scalar ◽

Partition Functions ◽

Parallel Plates ◽

Modular Transformations ◽

Real Analytic ◽

Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

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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.

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Metaplectic flavor symmetries from magnetized tori

Journal of High Energy Physics ◽

10.1007/jhep05(2021)078 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Yahya Almumin ◽

Mu-Chun Chen ◽

Víctor Knapp-Pérez ◽

Saúl Ramos-Sánchez ◽

Michael Ratz ◽

...

Keyword(s):

Zero Mode ◽

Metaplectic Groups ◽

Common Multiple ◽

Least Common Multiple ◽

Yukawa Couplings ◽

Flavor Symmetries ◽

Euler’S Theorem ◽

Modular Transformations ◽

Analytic Expressions ◽

Geometric Picture

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

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