Covering sumsets of a prime field and class numbersAlkan, Emre
2023 Research in the Mathematical Sciences
doi: 10.1007/s40687-023-00404-z
We study covering sumsets of a prime field based on its multiplicative structure. By developing various sufficient analytic and algebraic criteria for their existence, it is shown that covering sumsets arise in two main families, namely in the form of complementary sumsets and in the form of double sumsets. In each case, the abundance of covering sumsets is supported by providing asymptotically growing lower bounds on their number which in turn point out a rich array of fruitful connections to seemingly unrelated topics such as the Titchmarsh divisor problem, Mersenne primes, Fermat quotients, partitions into cycles, quadratic reciprocity, Gauss and Jacobi sums, and density results in class field theory resulting from Chebotarev’s theorem. Moreover, representations of an element taken from a prime field, in terms of the sums in a covering sumset, furnish us with new formulas for the class numbers of quadratic fields, Bernoulli numbers and Bernoulli polynomials. In this way, curious tendencies among the number of representations are discovered over half intervals. Lastly, our findings show in different circumstances that the summands of a covering sumset can seldom form an arithmetic progression, thereby indicating a tension between additive and multiplicative structures in a prime field.
Twisted Neumann–Zagier matricesGaroufalidis, Stavros; Yoon, Seokbeom
2023 Research in the Mathematical Sciences
doi: 10.1007/s40687-023-00400-3
The Neumann–Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation. They can be used as input data for the construction of a number of quantum invariants that include the loop invariants, the 3D-index and state-integrals. We define a twisted version of Neumann–Zagier matrices, describe their symplectic properties, and show how to compute them from the combinatorics of an ideal triangulation. As a sample application, we use them to define a twisted version of the 1-loop invariant (a topological invariant) which determines the 1-loop invariant of the cyclic covers of a hyperbolic knot complement, and conjecturally is equal to the adjoint twisted Alexander polynomial.
Explicit transformations for generalized Lambert series associated with the divisor function σa(N)(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{a}^{(N)}(n)$$\end{document} and their applicationsBanerjee, Soumyarup; Dixit, Atul; Gupta, Shivajee
2023 Research in the Mathematical Sciences
doi: 10.1007/s40687-023-00401-2
Let σa(N)(n)=∑dN|nda\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _a^{(N)}(n)=\sum _{d^{N}|n}d^a$$\end{document}. An explicit transformation is obtained for the generalized Lambert series ∑n=1∞σa(N)(n)e-ny\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum _{n=1}^{\infty }\sigma _{a}^{(N)}(n)e^{-ny}$$\end{document} for Re(a)>-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {Re}(a)>-1$$\end{document} using the recently established Voronoï summation formula for σa(N)(n)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _a^{(N)}(n)$$\end{document} and is extended to a wider region by analytic continuation. For N=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N=1$$\end{document}, this Lambert series plays an important role in string theory scattering amplitudes as can be seen in the recent work of Dorigoni and Kleinschmidt. These transformations exhibit several identities—a new generalization of Ramanujan’s formula for ζ(2m+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\zeta (2m+1)$$\end{document}, an identity associated with extended higher Herglotz functions, generalized Dedekind eta transformation, Wigert’s transformation, etc., all of which are derived in this paper, thus leading to their uniform proofs. A special case of one of these explicit transformations naturally leads us to consider generalized power partitions with “n2N-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n^{2N-1}$$\end{document} copies of nN\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n^{N}$$\end{document}.” Asymptotic expansion of their generating function as q→1-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$q\rightarrow 1^{-}$$\end{document} is also derived which generalizes Wright’s result on the plane partition generating function. In order to obtain these transformations, several new intermediate results are required, for example, a new reduction formula for Meijer G-function and an almost closed-form evaluation of ∂∂βE2N,β(z2N)β=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left. \frac{\partial }{\partial \beta }E_{2N, \beta }(z^{2N})\right| _{\beta =1}$$\end{document}, where Eα,β(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{\alpha , \beta }(z)$$\end{document} is the two-variable Mittag–Leffler function.