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Higher dimensional dedekind sums
Mathematische Annalen, 1973 In this paper we will study the number-theoretical properties of the expression v1 nkal rcka,, d(p; a I . . . . . an) = ( 1) n/2 ~ cot cot (1) k=l P P and of related finite trigonometric sums. In Eq. (I), p is a positive integer, a~ . . . . . a, are integers prime to p, and n is even (for n odd the sum is clearly equal to zero).
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Dedekind sums for a fuchsian group, II [PDF]
Nagoya Mathematical Journal, 1973 In [1] we derived a generalization of Kronecker’s first limit formula. Our generalization was a limit formula for the Eisenstein series for an arbitrary cusp of a Fuchsian group Γ of the first kind operating on the complex upper half-plane H. In that work, we introduced Dedekind sums associated to the principal congruence subgroups Γ(N) of the elliptic
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Conservation Science and Practice, Volume 7, Issue 12, December 2025.This study introduces a method to integrate extreme climate event indices into biodiversity assessment strategies, using South Africa as a case study. By assessing Key Biodiversity Areas (KBAs) and extreme climate projections from 2015 to 2036, the findings highlight significant gaps between areas officially designated as climate‐threatened and those ...
Amina Ly, Noah S. Diffenbaugh
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Generalized Dedekind sums and equidistribution mod 1
, 2015 Dedekind sums are well-studied arithmetic sums, with values uniformly distributed on the unit interval. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of $SL(2,\mathbb{Z})$.
Burrin, Claire
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Asymptotic Formulas and Generalized Dedekind Sums [PDF]
Experimental Mathematics, 1998 We find asymptotic formulas as $n\to\infty$ for the coefficients $a(r\hbox{,}\,n)$ defined by $$ \prod_{\nu=1}^\infty\,(1-x^\nu)^{-\nu^r} =\sum_{n=0}^\infty a(r\hbox{,}\,n)x^n\hbox{.} $$ (The case $r=1$ gives the number of plane partitions of $n$.) Generalized Dedekind sums occur naturally and are studied using the Finite Fourier Transform. The methods
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Chebotarev's theorem for cyclic groups of order pq$pq$ and an uncertainty principle
Bulletin of the London Mathematical Society, Volume 57, Issue 12, Page 3841-3856, December 2025.Abstract Let p$p$ be a prime number and ζp$\zeta _p$ a primitive p$p$th root of unity. Chebotarev's theorem states that every square submatrix of the p×p$p \times p$ matrix (ζpij)i,j=0p−1$(\zeta _p^{ij})_{i,j=0}^{p-1}$ is nonsingular. In this paper, we prove the same for principal submatrices of (ζnij)i,j=0n−1$(\zeta _n^{ij})_{i,j=0}^{n-1}$, when n=pr ...
Maria Loukaki
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The shift‐homological spectrum and parametrising kernels of rank functions
Journal of the London Mathematical Society, Volume 112, Issue 6, December 2025.Abstract For any compactly generated triangulated category, we introduce two topological spaces, the shift spectrum and the shift‐homological spectrum. We use them to parametrise a family of thick subcategories of the compact objects, which we call radical.
Isaac Bird +2 more
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Periodic points of rational functions over finite fields
Transactions of the London Mathematical Society, Volume 12, Issue 1, December 2025.Abstract For q$q$ a prime power and ϕ$\phi$ a rational function with coefficients in Fq$\mathbb {F}_q$, let p(q,ϕ)$p(q,\phi)$ be the proportion of P1Fq$\mathbb {P}^1\left(\mathbb {F}_q\right)$ that is periodic with respect to ϕ$\phi$. Furthermore, if d$d$ is a positive integer, let Qd$Q_d$ be the set of prime powers coprime to d!$d!$ and let P(d,q ...
Derek Garton
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Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums. [PDF]
Math Ann, 2023 Minelli P, Sourmelidis A, Technau M.
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On the Mean Value of Dedekind Sums
Journal of Number Theory, 2001 For integer \(m\geq 1\) define \(f_m(k)\) be the Dirichlet series \[ \sum^\infty_{k=1} {f_m(k)\over k^s}=2 {\zeta^2(2m) \over\zeta (4m)}{\zeta (s+4m-1)\over \zeta^2(s+2m)} \zeta(s), \] and let \(s(h,k)\) denote the classical Dedekind sum. If \(\sum'\) indicates that \((h,k)=1\), the author derives the asymptotic formula \[ {\sum'}^k_{h=1} s^{2m}(h,k ...
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