Related papers: Large Values of Newform Dedekind Sums

We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in $(0,1]$ ordered by denominators. We show convergence to a stable law…

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…

In this paper we prove a number of theorems that determine the extent to which the signs of the Hecke eigenvalues of an Eisenstein newform determine the newform. We address this problem broadly and provide theorems of both individual and…

Consider the partially ordered set on $[t]^n:=\{0,\dots,t-1\}^n$ equipped with the natural coordinate-wise ordering. Let $A(t,n)$ denote the number of antichains of this poset. The quantity $A(t,n)$ has a number of combinatorial…

We obtain a new motivated proof of the reciprocity law for Dedekind sums by computing the constant coefficient of the Ehrhart polynomial for a rectangular triangle in two ways. On the one hand, the constant term is the Euler characteristic,…

A natural question about Dedekind sums is to find conditions on the integers $a_1, a_2$, and $b$ such that $s(a_1,b) = s(a_2, b)$. We prove that if the former equality holds then $ b \ | \ (a_1a_2-1)(a_1-a_2)$. Surprisingly, to the best of…

Limit laws for ergodic averages with a power singularity over circle rotations were first proved by Sinai and Ulcigrai, as well as Dolgopyat and Fayad. In this paper, we prove limit laws with an estimate for the rate of convergence for the…

We prove the quantum modularity of the signature of $ \mathrm{SU}(2) $-TQFT for a genus 2 surface, which was conjectured by March\'{e}--Masbaum in 2025. Our approach is based on a quantum modularity of generalized Dedekind sums associated…

We show that the graph of normalized elliptic Dedekind sums is dense in its image for arbitrary imaginary quadratic fields, generalizing a result of Ito in the Euclidean case. We also derive some basic properties of Martin's continued…

It is proved that \[ \sum_{\chi \bmod q}N(\sigma , T, \chi) \lesssim_{\epsilon} (qT)^{7(1-\sigma)/3+\epsilon}, \] where $N(\sigma, T, \chi)$ denote the number of zeros $\rho = \beta + it$ of $L(s, \chi)$ in the rectangle $\sigma \leq \beta…

Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\{1,2,3,5,\dots\}$. This has been extended to many other recurrence relations $\{G_n\}$ (with their own notion of a legal…

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)=\sum_{x\leq…

Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for…

We prove a first Kronecker limit formula for cofinite discrete subgroups of SL$(2,\mathbb{C})$, also called Kleinian groups, generalizing a method of Goldstein over SL$(2,\mathbb R)$. The proof uses the Fourier expansion of Eisenstein…

We study reciprocity formulas for Dedekind sums associated with absolutely continuous functions, extending the classical Dedekind-Rademacher reciprocity formula. In particular, we treat the case of periodic Bernoulli functions. Our approach…

Let $f(z)=\sum_{n=1}^{\infty}a(n) e^{2\pi i nz}$ be a normalized Hecke eigenform in $S_{2k}^{\text{new}}(\Gamma_0(N))$ with integer Fourier coefficients. We prove that there exists a constant $C(f)>0$ such that any integer is a sum of at…

Many exponential sums over finite fields, including Gauss sums and Kloosterman sums, arise as the Fourier transform with respect to a character of the trace function of an $\ell$-adic sheaf on a commutative algebraic group. We study the…

We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…

Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Recently, Du and Zhang proved the following reciprocity formula. If $a$ and $b$ are odd natural numbers, $(a,b)=1$, then $$ S(2a^*,b)+S(2b^*,a)=\frac{a^2+b^2+4}{2ab}-3,…

We investigate the averages of Dedekind sums over rational numbers in the set $$\mathscr{F}_\alpha(Q):=\{\, {v}/{w}\in \mathbb{Q}: 0<w\leq Q\,\}\cap [0, \alpha)$$ for fixed $\alpha\leq 1/2$. In previous work, we obtained asymptotics for…