In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain tuples of real quadratic fields. At the end, we give an application of this result to produce some elliptic curves having a $3$-torsion subgroup.
It is expected that Schanuel's Conjecture contains all ``reasonable" statements that can be made on the values of {\em the exponential function}. In particular it implies the Lindemann-Weierstrass Theorem and the Conjecture on algebraic independence of logarithms of algebraic numbers. Our goal is to state conjectures {\em à la Schanuel}, which imply conjectures {\em à la Lindemann-Weierstrass}, for the exponential map of an extension $G$ of an elliptic curve ${\mathcal E}$ by the multiplicative group ${\mathbb G}_m$. In the present paper we assume that the extension is split, that is $G={\mathbb G}_m\times {\mathcal E}$. In a second paper in preparation we will deal with the non-split case, namely when the extension is not a product. Here we propose the {\em split semi-elliptic Conjecture}, which involves the exponential function and the Weierstrass $\wp$ and $ζ$ functions, related with integrals of the first and second kind. In the second paper, our {\em non-split semi-elliptic Conjecture} will also involve Serre's functions, related with integrals of the third kind. We expect that our conjectures contain all ``reasonable" statements that can be made on the values of these functions. In the present paper we highlight the geometric origin of the split semi-elliptic Conjecture: it is {\em equivalent to} the Grothendieck-André generalized period Conjecture applied to the 1-motive $M=[u:\mathbb{Z} \rightarrow {\mathbb G}_m^s \times {\mathcal E}^n ]$, which is […]
The study of finding blocks of primes in certain arithmetic sequences is one of the classical problems in number theory. It is also very interesting to study blocks of consecutive elements in such sequences that are pairwise coprime. In this context, we show that if $f$ is a twice continuously differentiable real-valued function on $[1, \infty)$ such that $f''(x) \to 0$ as $x \to \infty$, then there exist arbitrarily long blocks of pairwise coprime consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ if and only if $f'$ is unbounded. Among other, this result extends the qualitative part of a recent result by the first author, Drmota and Müllner.We also prove that, under the same conditions, there exists a subset $\mathcal{A} \subseteq \mathbb{N}$ having upper Banach density one such that for any two distinct integers $m, n \in \mathcal{A}$, the integers $\lfloor f(m) \rfloor$ and $\lfloor f(n) \rfloor$ are pairwise coprime. Further, we show that there exist arbitrarily long blocks of consecutive elements in the sequence $(\lfloor f(n) \rfloor)_n$ such that no two of them are coprime.
Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function $\mathcal{F}(z;u,v)$, defined as \begin{align*}\mathcal{F}(z;u,v) = \int_{0}^{1} \frac{\log(1-ut^z)}{v^{-1}-t} dt, \quad \mathrm{for} \,\,\,\, \mathfrak{Re}(z)>0.\end{align*}They obtained two-term, three-term and six-term functional equations for $\mathcal{F}(z;u,v)$ and also evaluated special values in terms of di-logarithmic functions. Motivated from their work, we study the following two integrals, \begin{align*}\mathcal{F}(z;u,v,w) &=\int_{0}^1 \frac{\log(1-ut^z)\log(1-wt^z)}{v^{-1}-t}\text{d}t, \\\mathcal{F}_k(z;u,v) &= \int_{0}^{1} \frac{\log^k(1-ut^z)}{v^{-1}-t} \, \text{d}t,\end{align*}for $\mathfrak{Re}(z)>0$ and $k \in \mathbb{N}$. For $k=1$, the integral $\mathcal{F}_k(z;u,v)$ reduces to $\mathcal{F}(z;u,v)$. This allows us to recover the properties of $\mathcal{F}(z;u,v)$ by studying the properties of $\mathcal{F}_k(z;u,v)$. We evaluate special values of these two functions in terms of poly-logarithmic functions.
We evaluate some determinants involving harmonic numbers that are needed in order to provide solutions to certain matrix equations occurring in our earlier paper \cite{CG24}. That paper determined the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. The present paper computes these determinants with the aid of two computer packages.
For almost all tuples $(x_1,\dots,x_n)$ of complex numbers, a strong version of Schanuel's Conjecture is true: the $2n$ numbers $x_1,\dots,x_n, {\mathrm e}^{x_1},\dots, {\mathrm e}^{x_n}$ are algebraically independent. Similar statements hold when one replaces the exponential function ${\mathrm e}^z$ with algebraically independent functions. We give examples involving elliptic and quasi--elliptic functions, that we prove to be algebraically independent: $z$, $\wp(z)$, $ζ(z)$, $σ(z)$, exponential functions, and Serre functions related with integrals of the third kind.
Let $F_n$ denote the $n$\textsuperscript{th} Fibonacci number (with first few terms 1, 2, 3, 5). The Zeckendorf game starts with $N$ chips on pile $F_1$, and on a turn a player may do one of two moves: if there are chips on piles $F_k$ and $F_{k-1}$ they may remove those and place one on $F_{k+1}$, or if there are two chips on $F_k$ they may remove those and put one on $F_{k+1}$ and one on $F_{k-2}$ (unless $k \le 2$: if there are two chips on $F_2$ we replace that with one each on $F_3$ and $F_1$, while if there are two on $F_1$ we replace with one on $F_2$). Note at any moment in the game if $c_n$ is the number of chips on pile $F_n$ then $\sum c_n F_n = N$. Two players alternate turns until no moves remain. It was quickly proved that if $N > 2$ then Player 2 has a winning strategy and the game ends in the Zeckendorf decomposition of $N$ (i.e., the representation of $N$ as a sum of non-adjacent Fibonacci numbers); unfortunately it's a non-constructive proof. Subsequent papers proved many results about the game and generalizations, but the winning strategy remains a mystery. We thus consider a new variant: the Black Hole Zeckendorf game, where there is a fixed $m$ such that if a chip is placed on pile $F_m$ it becomes inaccessible (we also allow the two players to add chips to set up the initial board, so we do not have to start with $N$ chips on $F_1$). We constructively find winning strategies for some $m$, where the winner depends on $N \bmod 16$.
Using a recent method developed by Mahatab, we obtain an improved $\Omega$-bound for the error term arising in lattice counting problem of bodies of revolution in $\mathbb R^3$ around a coordinate axis and having smooth boundary with bounded nonzero curvature. This strengthens an earlier result by Kühleitner and Nowak.
In the present article, we obtain thirteen special cases of Ramanujan's continued fraction and establish $2$-, $4$-dissections of those continued fractions. Also, we derive algebraic relations connecting continued fractions with their respective reciprocals and we validate our results establishing the coloured partition identities.
Let $ \ell, k \ge 2 $ be integers. In this article, we investigate the $ \ell $th power moments of the arithmetic function $ \tau_{k,\mathbb{K}}(n) $, which counts the number of ways $ n $ can be expressed as a product of norms of $k$ non-zero ideals in the ring of integers of a non-normal cubic field $ \mathbb{K} $, evaluated over integers represented as sums of eight squares.
In the present paper we generate binary pseudorandom sequences using generalized polynomials. A generalized polynomial is a function in whose description we not only allow addition and product (as it is the case in usual polynomials) but also the floor function. We estimate the well-distribution measure, looking at the "randomness" along arithmetic progressions.
This article is a generalisation of the results of Thorner etal concerning Benford's law for the Fourier coefficients of a primitive i.e. a normalised newform $f$ of weight $k$ on $\Gamma_0 (N)$ without complex multiplication to the Fourier coefficients of ${\rm sym}^2f$, where ${\rm sym}^2 f$ is the symmetric square lift of $f$.
We give a combinatorial proof of the limit theorem for triple convolutions of Ramanujan sums that had been earlier proved by Chaubey, Goel and Murty using the theory of exponential sums. This result has applications to the study of triple convolutions of certain arithmetical functions.
Chapter 12 in Ramanujan's second notebook is devoted to continued fractions, many of which are evaluated in terms of Gamma functions. Using the theory of hypergeometric functions, we give new proofs of two of the most attractive continued fractions, and also two new companions.
In [1] a conjecture for the modular transformation of the free fermion generalised Gibbs ensemble (GGE) was given where only the KdV charge associated to the weight four quasi primary field was inserted. In this paper we first generalise this conjecture to the case with an arbitrary, finite collection of KdV charges in the GGE. These GGEs are generalisations of the generating function of power partitions. We prove the conjectured transformation for the case with a finite number of charges inserted and discuss the case with an infinite number of charges.
In the recent past, the work in the area of vanishing coefficients of infinite $q$-products has been taken to the forefront. Weaving the same thread as Ramanujan, Richmond, Szekeres, Andrews, Alladi, Gordon, Mc Laughlin, Baruah, Kaur, Tang, we further prove vanishing coefficients in arithmetic progressions moduli 5, 7, 11, 13, 17, 19 and 29 of the following four families of infinite products, where $\{X_{a,b,sm,km,u,v}(n)\}_{n\geq n_0}$, $\{Y_{a,b,sm,km,u,v}(n)\}_{n\geq n_0}$ and $\{Z_{a,b,sm,km,u,v}(n)\}_{n\geq n_0}$ are defined by\begin{align*}\sum_{n\geq n_0}^{\infty}X_{a,b,sm,km,u,v}(n)q^n:=&\p(q^{a},q^{sm-a};q^{sm})^u\p(q^{b},q^{km-b};q^{km})^v, \\\sum_{n\geq n_0}^{\infty}Y_{a,b,sm,km,u,v}(n)q^n:=&\p(q^{a},q^{sm-a};q^{sm})^u\p(-q^{b},-q^{km-b};q^{km})^v, \\\sum_{n\geq n_0}^{\infty}Z_{a,b,sm,km,u,v}(n)q^n:=&\p(-q^{a},-q^{sm-a};q^{sm})^u\p(q^{b},q^{km-b};q^{km})^v,\end{align*}here $a, b, s, k, u$ and $v$ are chosen in such a way that the infinite products in the right-hand side of the above are convergent and $n_0$ is an integer (possibly negative or zero) depending on $a, b, s, k, u$ and $v$. The proof uses the Jacobi triple product identity and the properties of Ramanujan general theta function.