21 problems

Let $s(m,n)$ be the classical Dedekind sum for coprime integers $m,n$ with $n\neq 0$, defined by … For a rational number $m/n$ in lowest terms, write $s(m/n):=s(m,n)$. Grosswald–Ra…

Let $\mathfrak{s}(h,k)$ denote the classical Dedekind sum, and average over reduced fractions $h/k$ with $1\leq h\leq k$. Existing bounds show that the normalized average of…

Let $g=2$ and let $x=r/p$ be a rational number with coprime odd integers $1\leq r<p$. Write $\sigma_2(x)\in\mathbb{Z}$ for the signature associated with the corresponding…

Let $\chi_1$ and $\chi_2$ be nontrivial, primitive Dirichlet characters modulo $q_1$ and $q_2$, respectively, with $\chi_1\chi_2(-1)=1$. Let $F_{\chi_1,\chi_2}$ be the smallest num…

Let $\chi_1$ and $\chi_2$ be nontrivial, primitive Dirichlet characters modulo $q_1$ and $q_2$, respectively, with $\chi_1\chi_2(-1)=1$. Let $S_{\chi_1,\chi_2}$ be their generalize…

Prime-modulus Dedekind-sum sign conjecture. (i) As $p$ ranges over the odd primes and $H$ over the subgroups of odd order of $({\mathbb Z}/p{\mathbb Z})^*$, the inequality

Girstmair's conjecture. The number $\frac{k}{q}$ is a value of the normalized Dedekind sum if, and only if, the following conditions hold: if $3\nmid q$, then $k\equiv0\pmod{3}$; a…

Order-sensitive Dedekind-sum conjecture. There exists an absolute constant $C>0$ such that

Square-root bound conjecture. One has

Affine-linearity conjecture. Then

Let $a$ be an even integer and $b$ an odd integer, and let $s_{\sqrt{-1}}(a;b)$ and $Q(a;b)$ denote the elliptic Dedekind sum and the associated quantity used in the paper. For an…

Let $q\geq 2$ and $r\geq 1$, and define $h(u,v)$ as above. The generalized one-parameter range conjecture. If $x$ ranges over all integers satisfying $\gcd(x+rq,xrq-1)=1$ and…

For $q\geq 2$, define $h(u,v)$ as in the two-parameter construction, and consider the specialization $u=x+q$, $v=xq-1$. The one-parameter range conjecture. If $x$ ranges over all i…

For $q\geq 2$ and coprime integers $x,y$ satisfying $\gcd(q,x^2+y^2)=1$, define $h(x,y)$ by … where $yy^{*}\equiv 1\pmod{x^2+y^2}$, and let $\Gamma_q$ be the set defined above. The…

For an integer $q\geq 2$, let $\Lambda_q$ be the set of residue classes produced by the auxiliary function $\lambda(a,t,t^{*})$, and define … The range conjecture. For every intege…

Let $S(a,b)$ be the normalized Dedekind sum, and write a value in lowest terms as $S(a,b)=k/q$, where $q\geq 2$ and $\gcd(k,q)=1$. Girstmair's necessary conditions are that…

Let $s(p,q)$ be the Dedekind sum … where $p,q$ are coprime integers with $q>0$, and $((x))$ is the sawtooth function, equal to $x-[x]-\frac12$ for $x\notin\mathbb{Z}$ and to $0$ fo…

For a fixed positive integer $b$, let $\operatorname{inv}(a,b)$ denote the inversion number and let $I_n(b)$ be its $n$th smallest value, where $1\leq n\leq6$. Six-smallest-values…

Let $b=ck$, and let $\zeta_m$ denote a primitive $m$th root of unity. The inversion polynomial $f_b(x)$ is defined by its roots indexed by the inversion numbers $\inv(a,b)$. Cyclot…

Recursive-algorithm conjecture. There exist recursive Euclidean-like algorithms for evaluating the sums associated with permutation polynomials of general degree $p$, analogous to…

The Dedekind-sum parity conjecture. If $p\geq 5$ is prime and $(k,p)=1$, then