If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a+b, where a\in A and b\in B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet-Thue-Morse sequence, {0} \cup {2^n : n \geq 0}, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that have an odd number of partitions, then B is the set of pentagonal numbers {n(3n+1)/2 : n \in Z}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.
We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan-Fibonacci relations. Then we focus on the general properties of the family and prove explicit formulas, including two types of integral representations.
Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}ε_i2^i$, where $ε_i\in\mathcal{A}$. The sequence $\left(f_\mathcal{A}(n)\right) \bmod 2$ is always periodic, and $f_\mathcal{A}(n)$ is typically more often even than odd. We give four families of sets $\left(\mathcal{A}_m\right)$ with $\left|\mathcal{A}_m\right|=4$ such that the proportion of odd $f_{\mathcal{A}_m}(n)$'s goes to $1$ as $m\to\infty$.
Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.
We obtain results bounding the degree of the series $\sum_{n=1}^{\infty} 1/α_n$, where $\{α_n\}$ is a sequence of algebraic integers satisfying certain algebraic conditions and growth conditions. Our results extend results of Erdős, Hančl and Nair.
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.
In this paper we give a classification of the asymptotic expansion of the $q$-expansion of reciprocals of Eisenstein series $E_k$ of weight $k$ for the modular group $\func{SL}_2(\mathbb{Z})$. For $k \geq 12$ even, this extends results of Hardy and Ramanujan, and Berndt, Bialek and Yee, utilizing the Circle Method on the one hand, and results of Petersson, and Bringmann and Kane, developing a theory of meromorphic Poincar{é} series on the other. We follow a uniform approach, based on the zeros of the Eisenstein series with the largest imaginary part. These special zeros provide information on …
Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases, for n restricted to certain arithmetic progressions, it is true. We prove such zero-density results using the theory of modular forms, and speculate about what may be true in general.
Using the reflection formula of the Gamma function, we derive a new formula for the Taylor coefficients of the reciprocal Gamma function. The new formula provides effective asymptotic values for the coefficients even for very small values of the indices. Both the sign oscillations and the leading order of growth are given.
In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.
Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known convergence acceleration for the ordinary prime zeta functions, the sums and first derivatives are tabulated with high precision for indices k=2,...,6 and integer powers s=2,...,8.
Let $X$ be a set of positive integers, and let $\mathbb Z_K$ be the ring of integers of a number field $K$ of degree $n$. Denote by $N(I)$ the absolute norm of an ideal $I$ of $\mathbb Z_K$, and by $\mathcal A$ the set of principal ideals $a\mathbb Z_K$ such that $a$ is an atom of $\mathbb Z_K$ and $a$ divides $m$ for some $m \in X$. Building upon the ideas of Clarkson from [Proc. Amer. Math. Soc. 17 (1966), 541], we show that, if the series $\sum_{m \in X} 1/m$ diverges, then so does the series $\sum_{\mathfrak a \in \mathcal A} |N(\mathfrak a)|^{-1/n}$. Most notably, this generalizes a class…
This translation has been published in Stephen Hawking (ed.), "God Created the Integers", published in 2007 by Running Press. There may have been some changes to the final published version and this copy. This is a translation from the Latin original, "De summis serierum reciprocarum" (1735). E41 in the Enestrom index. In this paper Euler finds an exact expression for the sum of the squares of the reciprocals of the positive integers, namely pi^2/6. He shows this by applying Newton's identities relating the roots and coefficients of polynomials to the power series of the sine function. Indeed,…
We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty}.$$ If \begin{align*} \sum_{n=0}^{\infty}α(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}α^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}β(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{…
We study certain series with Catalan numbers and reciprocal Catalan numbers, respectively, and provide seemingly new closed form evaluations of these series with Fibonacci (Lucas) entries. In addition, we state some combinatorial sums that can be inferred from the series.
If \(A \) is a set of natural numbers containing \(0 \), then there is a unique nonempty "reciprocal" set \(B \) of natural numbers (containing \(0 \)) such that every positive integer can be written in the form \(a + b \), where \(a \in A \) and \(b \in B \), in an even number of ways. Furthermore, the generating functions for \(A \) and \(B \) over \(\FF_2 \) are reciprocals in \(\FF_2 [[q]] \). We consider the reciprocal set \(B \) for the set \(A \) containing \(0 \) and all integers such that \(σ(n) \) is odd, where \(σ(n) \) is the sum of all the positive divisors of \(n \). This problem…
The paper studies algebraic independence of certain reciprocal sums of Fibonacci and Lucas sequences. Also more general binary recurrences are considered. The main tool is Mahler's method reducing the investigation of the algebraic independence of function values to the one of functions if these satisfy certain types of functional equations.
It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are known to be all convergent. Here, letting $S^{(r)}$ $(r \in \mathbb{N})$ denote the series of the reciprocal of the positive integers whose the decimal represent…
We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.
We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight $3/2$ whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla-Millson theta lift of Bruinier and Funke to meromorphic modular functions.
In this paper, we prove the following result: {quote} Let $\A$ be an infinite set of positive integers. For all positive integer $n$, let $τ_n$ denote the smallest element of $\A$ which does not divide $n$. Then we have $$\lim_{N \to + \infty} \frac{1}{N} \sum_{n = 1}^{N} τ_n = \sum_{n = 0}^{\infty} \frac{1}{\lcm\{a \in \A | a \leq n\}} .$${quote} In the two particular cases when $\A$ is the set of all positive integers and when $\A$ is the set of the prime numbers, we give a more precise result for the average asymptotic behavior of ${(τ_n)}_n$. Furthermore, we discuss the irrationality of th…
Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erdös and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\geq c>2$. The bound $c>2$ is best possible thanks to…
We consider the p-adic distributions derived from Eisenstein series in [GMPS], whose Mellin transforms are reciprocals of the Kubota-Leopoldt p-adic L-function. These distributions were shown there to be measures when p is regular. They fail to be measures when p is irregular; in this paper we give quantitative estimates that describe their behavior more precisely.
It is known that the sum of the reciprocal of integers, $\sum_n (1/n)$, and the sum of the reciprocal of primes, $\sum_n (1/p_n)$, both diverge. Here, we study a series made from primes that sums exactly to 1. We also show this sum is simply related to an infinite product over primes. We then generalize the form of the series (and its related product), extending this idea to other number theoretic sets such as twin primes. Evaluating a product made from twin primes gives a new mathematical constant.
The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.
For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property that partial sums of the series $\{\frac{1}{n_i}\}_{i=1}^{k}$ can only represent the integers with the form $\sum_{i\in I}m_i$, where $I\subset\{1,...,e\}$.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent harmonic series.
A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with both convergent and divergent harmonic series.
A cubic (resp. biquadratic) theta series is a power series whose n-th coefficient is equal to 1 if n is a perfect cube (resp. fourth power) and zero otherwise. We improve on a result of Bradshaw by showing that such series is not a cubic (resp. biquadratic) algebraic number when evaluated at reciprocals of integers. The proof relies on a "nested gaps technique" for linear independence and on recent results by the author on Waring's problem for cubes and biquadrates.
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fib…
For certain sequences $A$ of positive integers with missing $g$-adic digits, the Dirichlet series $F_A(s) = \sum_{a\in A} a^{-s}$ has abscissa of convergence $σ_c < 1$. The number $σ_c$ is computed. This generalizes and strengthens a classical theorem of Kempner on the convergence of the sum of the reciprocals of a sequence of integers with missing decimal digits.
In this note we present a method to bound gaps between primes via the divergence of the series of reciprocals of the prime numbers, a consequence of a version of the Bertrand's test for convergence of series of positive numbers and a suitable series of positive terms related to the logarithmic integral function. Using this proposed method we discuss the Polignac's conjecture and present some consequences.
We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as $q$-series identities of the mock theta type.
In this paper we investigate a result of Ueno on the modularity of generating series associated to the zeta functions of binary Hermitian forms previously studied by Elstrodt et al. We improve his result by showing that the generating series are Eisenstein series. As a consequence we obtain an explicit formula for the special values of zeta functions associated with binary Hermitian forms.
We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant lattices, and investigate explicit relationships between Dirichlet series associated with those lattices. We also study the analytic properties of the Dirichlet series, and rewrite the functional equation in a self dual form using the explicit relationship.
As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic forms, the associated Dirichlet series satisfies a simple explicit relation to that of the dual other than the usual functional equation. As an application, we write the functional equations of these Dirichlet series in self dual forms.
An Engel series is a sum of the reciprocals of an increasing sequence of positive integers, which is such that each term is divisible by the previous one. Here we consider a particular class of Engel series, for which each term of the sequence is divisible by the square of the preceding one, and find an explicit expression for the continued fraction expansion of the sum of a generic series of this kind. As a special case, this includes certain series whose continued fraction expansion was found by Shallit. A family of examples generated by nonlinear recurrences with the Laurent property is con…
We state and prove an identity which connects theta series associated with binary quadratic forms of idoneal discriminants $Δ$ and $Δp^2$, for $p$ a prime. Employing this identity, we extend the results of Toh by writing the theta series of forms of discriminant $Δp^2$ as a linear combination of Lambert series. We then use these Lambert series decompositions to give explicit representation formulas for the forms of discriminant $Δp^2$. Lastly, we give a generalization of our main identity, which employs a map of Buell to connect forms of discriminant $Δ$ to $Δp^2$. Our generalized identity lin…
When the discriminants $Δ$ and $Δp^2$ are idoneal, Patane proved a theorem which connects the theta series associated to binary quadratic forms of each discriminant. This paper generalizes the main theorem of Patane by no longer requiring $Δ$ and $Δp^2$ to be idoneal. In particular, we state and prove an identity which connects the theta series associated to a single binary quadratic form of discriminant $Δ$ to a theta series associated to a subset of binary quadratic forms of discriminant $Δp^2$. Here and everywhere $p$ is a prime.
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as (reciprocal) logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms are given for Euler's constant and the constant ln(2*pi), and yet another generalization of Euler's constant is proposed and various formulas for …
