The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.
For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $aP_5(x)+2^rbP_5(y)+2^scP_5(z)$ represents all but finitely many positive integers.
For an integer $x$, an integer of the form $P_5(x)=\frac{3x^2-x}2$ is called a generalized pentagonal number. For positive integers $α_1,\dots,α_k$, a sum $Φ_{α_1,\dots,α_k}(x_1,x_2,\dots,x_k)=α_1P_5(x_1)+α_2P_5(x_2)+\cdots+α_kP_5(x_k)$ of generalized pentagonal numbers is called universal if $Φ_{α_1,\dots,α_k}(x_1,x_2,\dots,x_k)=N$ has an integer solution $(x_1,x_2,\dots,x_k) \in \mathbb Z^k$ for any non-negative integer $N$. In this article, we prove that there are exactly $234$ proper universal sums of generalized pentagonal numbers. Furthermore, the "pentagonal theorem of $109$" is proved,…
On a product of three theta functions and the number of representations of integers as mixed ternary sums involving squares, triangular, pentagonal and octagonal numbers
For any $m\ge3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ be positive integers. For every non negative integer $n$, if there are integers $x,y,z$ such that $n=ap_k(x)+bp_k(y)+cp_k(z)$, then the quadruple $(k;a,b,c)$ is said to be {\it universal}. Sun gave in \cite{s1} all possible quadruple candidates that are universal and proved some quadruples to be universal (see also \cite{gs}). He remains the following quadruples $(5,1,1,k)$ for $k=6,8,9,10$, $(5,1,2,8)$, and $(5,1,3,s)$ for $7 \le s \le 8$ …
In this article, we study the representability of integers as sums of pentagonal numbers, where a pentagonal number is an integer of the form $P_5(x)=\frac{3x^2-x}{2}$ for some non-negative integer $x$. In particular, we prove the "pentagonal theorem of $63$", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $11$, $13$, $14$, $17$, $18$, $19$, $23$, $28$, $31$, $33$, $34$, $39$, $42$, and $63$. We also introduce a method to obtain a generalized version of Cauchy's lemma using re…
In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form $\frac{3n^2 \pm n}{2}$. He recalls that the infinite product $(1-x)(1-x^2)(1-x^3)...$ expands into an infinite series with exponents the pentagonal numbers, and tries substituting the roots of this infinite product into this infinite series. I am not sure what he is doing in some parts: in particular, he does some complicated calculations about the roots of unity and sums of them, their squares, reciprocals, etc., and also sums some divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreci…
Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes the pentagonal number theorem and uses it to prove a recurrence relation for the sum of divisors function. The term "pentagonal numbers" comes from polygonal numbers. Euler takes the logarithmic derivative of both sides. Then after multiplying both sides by $-x$, the left side is equal to $\sum_{n=1}^\infty σ(n) x^n$, where $σ(n)$ is the sum of the divisors …
Given a positive linear combination of five (respectively seven) cosines, where the angles are positive and sum to pi, the aim of this article is to express the sharp bound of the combination as a Positive Real Fraction in the coefficients (hence cosine-free). The method uses algebraic and arithmetic manipulations with judicious transformations.
Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved a recurrence relation for the sum of divisors function. In E244 Euler proves the pentagonal number theorem, and then repeats the proof of the recurrence. In Proposition 1, Euler proves that $\prod_{n=1}^\infty (1-a_n)=1-\sum_{n=1}^\infty a_n(1-a_1)...(1-a_{n-1})$. In Proposition 2 he explicitly states the case $a_n=x^n$. Then in Proposition 3, Euler applies…
We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann Hypothesis(GRH) holds. As applications, we find all integers, under the GRH only when $k=3$, which are sums of $k$ nonzero triangular numbers, sums of $k$ nonzero generalized pentagonal numbers, and sums of $k$ nonzero generalized octagonal numbers, respectively for any integer $k\ge 3$.
A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the coefficients result from a discrete integration of Euler's coefficients. Both a bijective proof and one based on generating functions show the equivalence of the subject recurrences.
Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.
Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated serie…
The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised …
In this paper we shall evaluate two alternating sums of binomial coefficients by a combinatorial argument. Moreover, by combining the same combinatorial idea with partition theoretic techniques, we provide $q$-analogues involving the $q$-binomial coefficients.
In this paper, we mainly show that generalized hyperharmonic number sums with reciprocal binomial coefficients can be expressed in terms of classical (alternating) Euler sums, zeta values and generalized (alternating) harmonic numbers.
This paper contains a number of series whose coefficients are products of central binomial coefficients & harmonic numbers. An elegant sum involving $ζ(2)$ and two other nice sums appear in the last section.
In this article, we consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge10$, those weighted sums of generalized $m$-gonal numbers represent every non-negative integers if they only represent $1$, $m-4$, and $m-2$. Furthermore, we study representations of sums of four generalized polygonal numbers with coefficients $1$ or $2$.
We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$, and as an application obtain a pointwise estimate and a second moment estimate for the sums in question.
Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.
We derive explicit formulas for some Kloosterman sums on $Γ_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.
We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global principle.
We study the behavior of the shifted convolution sum involving fourth power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the $k$-full kernel function for any fixed integer $k\geq2$.
Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also conjectured by Guo. Our proof uses Ni and Pan's technique and a simple $q$-congruence observed by Guo and Schlosser.
Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.
We study sums of Hecke eigenvalues of Hecke-Maass cusp forms for the group $\mathrm{SL}(n,\mathbb Z)$, with general $n\geq 3$, over certain short intervals under the assumption of the generalised Lindelöf hypothesis and a slightly stronger upper bound concerning the exponent towards the Ramanujan-Petersson conjecture that is currently known. In particular, in this case we evaluate the second moment of the sums in question asymptotically.
We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL(2) holomorphic cusp forms, GL(2) Maass forms, and GL(3) Maass forms. Building on previous works by the authors, we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL(2) holomorphic cusp forms and Maass fo…
We estimate the mean square of a short exponential sum involving Fourier coefficients of a cusp form with a linear twist, a smooth weight function, and a relatively short averaging interval.
We investigate the coefficients of the polynomial \[ S_{m,r}^n(\ell)=r^n+(m+r)^n+(2m+r)^n+\cdots+((\ell-1)m+r)^n. \] We prove that these can be given in terms of Stirling numbers of the first kind and $r$-Whitney numbers of the second kind. Moreover, we prove a necessary and sufficient condition for the integrity of these coefficients.
We study the well-known lacunary sums of binomial coefficients considered, most notably, by Christian Ramus, and their connection to a special kind of harmonic number associated with the first case of Fermat's Last Theorem. For one case of Ramus's famous identity we obtain a variation in which some of the parameters are replaced by square matrices of arbitrary dimension.
The series of some new estimates for the sums of the type \[ S_{q}(x;f)\,=\,\mathop{{\sum}'}\limits_{n\leqslant x}f(n)e_{q}(an^{*}+bn) \] is obtained. Here $q$ is a sufficiently large integer, $\sqrt{q}(\log{q})\!\ll\!x\leqslant q$, $a,b$ are integers, $(a,q)=1$, $e_{q}(v) = e^{2πiv/q}$, $f(n)$ is a multiplicative function, $nn^{*}\equiv 1 \pmod{q}$ and the prime sign means that $(n,q)=1$.
In this paper, we investigate two methods to express the natural powers of $2$ as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiation of their generating function. Some experimental results suggest the existence of other methods of decomposing the power of $2$ as sums over partitions.
The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.
Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to an unexpected cancellation of oscillating sums. After stating the motivation, and our theorem, we apply it to prove several results on the Prouhet-Tarry-Escott Problem, integer partitions, and the distribution of prime numbers. Regarding the Prouhet-Tarry-Escott problem, we show that \begin{align*} \sum_{|\ell|\leq x}(4x^2-4\ell^2)^{2r}-\sum_{|\ell|<x}(4x^2-(2\ell+1)^2)^{2r}=\text{polynomial w.r.t. } x \text{ with degree }2r-1. \end{align*} This can perhaps be …
We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously …
We study sign changes in the sequence $\{ A(n) : n = c^2 + d^2 \}$, where $A(n)$ are the coefficients of a holomorphic cuspidal Hecke eigenform. After proving a variant of an axiomatization for detecting and quantifying sign changes introduced by Meher and Murty, we show that there are at least $X^{\frac{1}{4} - ε}$ sign changes in each interval $[X, 2X]$ for $X \gg 1$. This improves to $X^{\frac{1}{2} - ε}$ many sign changes assuming the Generalized Lindelöf Hypothesis.
A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop new closed form representations of sums of cubic alternating harmonic numbers and reciprocal binomial coefficients. Some inter- esting (known or new) illustrative special cases as well as immediate consequences of the main results are also considered.
We obtain resonances for short exponential sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(n,\mathbb Z)$. This involves deriving asymptotics for the integrals appearing in the $\mathrm{GL}(n)$ Voronoi summation formula. As an application, we also prove an $Ω$-result for short sums of Fourier coefficients.
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have…
We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Salié sums.
We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers, shifted harmonic numbers and Riemann zeta function with positive integer arguments. In particular we investigate products of quadratic and cubic harmonic numbers and reciprocal parametric binomial coefficients. Some illustrative special cases as well as immediate consequences of the main results are also considered.
We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.
