Related papers: On a Conjecture about Sums Involving Farey Fractio…
Let Q -> B be a quadric fibration and T -> B a family of sextic du Val del Pezzo surfaces. Making use of the recent theory of noncommutative mixed motives, we establish a precise relation between the Schur-finiteness conjecture for Q, resp.…
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
The Farey sequence is the sequence of all rational numbers in the real unit interval, stratified by increasing denominators. A classical result by Hall says that its normalized gap distribution is the same as the distribution of the random…
We study a finite analog of a conjecture of Erd\"os on the sum of the squared multiplicities of the distances determined by an $n$-element point set. Our result is based on an estimate of the number of hinges in spectral graphs.
The Kaneko--Zagier conjecture describes a correspondence between finite multiple zeta values and symmetric multiple zeta values. Its refined version has been established by Jarossay, Rosen and Ono--Seki--Yamamoto. In this paper, we…
The left and right $q$-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and they gave a homological interpretation for left and right $q$-deformed rational numbers. In the present paper,…
We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.
The sum formula is one of the most well-known relations among multiple zeta values. This paper proves a conjecture of Kaneko predicting that an analogous formula holds for finite multiple zeta values.
By using Andrews's explicit formulae of the $q$-Fibonacci sequence introduced by Schur, we prove certain congruences of the $q$-Fibonacci sequence which relate the sequence with the original Fibonacci sequence. As a corollary, we show that…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants $ f(\mathfrak a_1) - f(\mathfrak a_2)$ based on their calculation in \cite{YZ}. Here $\mathfrak a_i$ belong two…
Let \[ \sum_{n=0}^{\infty}A(n)q^{n} := \frac{(q^{2};q^{5})_{\infty}^{5}(q^{3};q^{5})_{\infty}^{5}}{(q;q^{5})_{\infty}^{5}(q^{4};q^{5})_{\infty}^{5}}, \] \[ \sum_{n=0}^{\infty} B(n)q^{n} := \frac{(q;q^{5})_{\infty}^{5}…
In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the…
In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$.…
In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of…
In this paper we generalize some of our results from, `A note on Farey fractions with odd denominators' to subsets of Farey fractions consisting of fractions with denominators not divisible by a given prime. We also investigate the joint…
The convergence of a sequence of Cauchy sequences is conjectured; which if shown to be true, would prove the Riemann hypothesis by way of LeClair and Fran\c{c}a's transcendental equation criteria.
We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli.
We give a new proof of a_4\phi_3 summation due to G.E. Andrews and confirm another_4\phi_3 summation conjectured by him recently. Some variations of these two_4\phi_3 summations are also given.
Let $f_1,\dots,f_m$ be polynomials in $n$ variables with coefficients in a finite field $\mathbb{F}_q$. We estimate the number of points $\underline{x}$ in $\mathbb{F}_q^n$ such that each value $f_i(\underline{x})$ is a nonzero square in…