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In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Starks conjecture. We also discuss connections with 1-cocycles for the modular group $mathrm{PSL}(2,mathbb{Z})$.
The generalized Fibonacci sequences are sequences ${f_n}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t in mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan,
The modified Bernoulli numbers begin{equation*} B_{n}^{*} = sum_{r=0}^{n} binom{n+r}{2r} frac{B_{r}}{n+r}, quad n > 0 end{equation*} introduced by D. Zagier in 1998 were recently extended to the polynomial case by replacing $B_{r}$ by the Bernoulli p
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diopha
We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis
In this short, we study sums of the shape $sum_{nleqslant x}{f([x/n])}/{[x/n]},$ where $f$ is Euler totient function $varphi$, Dedekind function $Psi$, sum-of-divisors function $sigma$ or the alternating sum-of-divisors function $beta.$ We improve pr