q-analogues of the sums of powers of consecutive integers

Sums of powers of consecutive q-integers

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

A note on the sums of powers of consecutive q-integers

In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.

Perfect powers that are sums of consecutive cubes

Euler noted the relation $6^3=3^3+4^3+5^3$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular Stroeker determined all squares that can be written as a sum of at most $50$ consecutive cubes. We generalize Stroeker's work by determining all perfect powers that are sums of at most $50$ consecutive cubes. Our methods include descent, linear forms in two logarithms, and Frey-Hellegouarch curves.

Integer ratios of consecutive alternating power sums

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$ \frac{1^k-2^k+3^k-\dots+(-1)^{n+1} n^k}{1^k-2^k+3^k-\dots+(-1)^{n}(n-1)^k} $$ of two consecutive alternating power sums is an integer.

Note on the sums of powers of consecutive $q$-integers

In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.

Lemke Oliver and Soundararajan bias for consecutive sums of two squares

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several …

A diophantine equation for sums of consecutive like powers

We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.

On powers that are sums of consecutive like powers

Let $k \ge 2$ be even, and let $r$ be a non-zero integer. We show that for almost all $d \ge 2$ (in the sense of natural density), the equation $$ x^k+(x+r)^k+\cdots+(x+(d-1)r)^k=y^n, \qquad x,~y,~n \in \mathbb{Z}, \qquad n \ge 2, $$ has no solutions.

两个平方数之和的连续序列的正密度

Positive density for consecutive runs of sums of two squares

本文研究了两个平方数之和的连续序列在等差数列中的分布，证明对于任意奇数无平方因子模数和简化同余类，存在正密度的此类连续序列，这是 Maynard 素数序列结果的类比。

A singular K3 surface related to sums of consecutive cubes

We study the surface arising from the diophantine equation $m^3+(m+1)^3+...+(m+k-1)^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function of the surface over $Q$.

算术多边形与连续平方和

Arithmetic Polygons and Sums of Consecutive Squares

本文研究了算术多边形与连续平方和的关系，证明了对于奇数N≥3存在至少一个N边算术多边形，并展示了存在无限多个偶数边的算术多边形。

The distribution of consecutive prime biases and sums of sawtooth random variables

In recent work, we considered the frequencies of patterns of consecutive primes $\pmod{q}$ and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to bot…

A Short Proof of a Known Relation for Consecutive Power Sums

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

Forbidden integer ratios of consecutive power sums

Let $S_k(m):=1^k+2^k+\cdots+(m-1)^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ the ratio $S_k(m+1)/S_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers $ρ$ as a ratio and combine them to exclude the integers $3\le ρ\le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $ρ$. To exclude a ratio $ρ$ one has to show that the Erdős-Moser type equation $(ρ-1)S_k(m)=m^k$ has no non-trivial solutions.

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

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.

On equal values of products and power sums of consecutive elements in an arithmetic progression

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.

有限域上连续平方的幂和，以及多项式不同值之和

On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials

本文研究有限域中连续元素为平方或非平方时的幂和，并将其用于计算四次多项式 X^4+aX^2+b 的像集元素之和，解决了两个已有猜想。

Determinations of Analogues of Gauss Sums and Other Trigonometric Sums

Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated with primitive odd characters. In this paper, we establish two general theorems involving both sines and cosines, with more attention given to cosine sums in the several examples that we provide.

Mean values of character sums analogue of Kloosterman sums

Let $q$ be a positive integer, $χ$ a nontrivial character mod $q$, $\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shall study the character sum analogue of the well-known Kloosterman sum,\[\sum_{\substack{a\in\mathcal{I} \gcd(a,q)=1}}χ(ma+n\overline{a}),\] where $\overline{a}$ is the multiplicative inverse of $a\bmod q$. The mean square values and bilinear forms for such sums are proved.

General solutions of sums of consecutive cubed integers equal to squared integers

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the difference and sum of the triangular numbers of $\left(a+M-1\right)$ and $\left(a-1\right)$ in the product of their greatest common divisor $g$ and remaining square factors $δ^{2}$ and $σ^{2}$, yielding $c=gδσ$. Further, the condition that $g$ must be integer for several particular and general cases yield generalized Pell equations whose solutions allow to find all …

On q-analogues of quadratic Euler sums

In this paper we define the generalized q-analogues of Euler sums and present a new family of identities for q-analogues of Euler sums by using the method of Jackson q-integral rep- resentations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we also use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative examples are considered. Finally, we can obtain some explicit relations for the classical Euler sums when q approaches to 1.

Sums of Consecutive Integers

A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by their lengths and introduce the concept of length spectrum. Also, we use diagrams to illustrate the ideas behind our proofs.

Superelliptic equations arising from sums of consecutive powers

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \ge 2$) was considered by Zhongfeng Zhang who solved it for $k=2$, $3$, $4$ using Frey-Hellegouarch curves and their Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for $k=5$ is $x=z=0$, and that there are no solutions for $k=6$. The chief innovation we employ is a computational one, which enables us to avoid the…

Congruent conditions on the number of terms, on the ratio number of terms to first terms and on the difference of first terms for sums of consecutive squared integers equal to squared integers

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$ when these are linked by $a_{1}+a_{2}=μM+1$ with $μ=\left(η/δ\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper, congruent conditions on $M,η,δ$, and on the difference $\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold. It is found that $η\equiv1\left(mod\,2\right)$ and $δ\equiv0,1$ or $5\left(mod\,6\right)$, and if $δ\equiv0\left(m…

Moments and equidistributions of multiplicative analogues of Kloosterman sums

We consider a family of character sums as multiplicative analogues of Kloosterman sums. Using Gauss sums, Jacobi sums and Deligne's bound for hyper-Kloosterman sums, we establish asymptotic formulae for any real (positive) moments of the above character sum as the character runs over all non-trivial multiplicative characters mod $p.$ Moreover, an arcsine law is also established as a consequence of the method of moments. The evaluations of these moments also allow us to obtain asymptotic formulae for moments of such character sums weighted by special $L$-values (at $1/2$ and $1$).

Euler's and the Taxi Cab relations and other numbers that can be written twice as sums of two cubed integers

We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.

Additional congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

The problem of finding all the integer solutions in $a$, $M$ and $s$ of sums of $M$ consecutive integer squares starting at $a^{2}\geq1$ equal to squared integers $s^{2}$, has no solutions if $M\equiv3,5,6,7,8$ or $10\left(mod\,12\right)$ and has integer solutions if $M\equiv0,9,24% or %33\left(mod\,72\right)$; or $M\equiv1,2$ or $16\left(mod\,24\right)$; or $M\equiv11\left(mod\,12\right)$. In this paper, additional congruence conditions are demonstrated on the allowed values of $M$ that yield solutions to the problem by using Beeckmans' eight necessary conditions, refining further the possibl…

On consecutive sums in permutations

We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high probability there are $(\frac{1+e^{-2}}{4} +o(1)) n^2$ such sums. This answers an old question of Erdős and Harzheim. We also obtain non-trivial bounds on the maximum possible number of distinct sums, ranging over all permutations of $1,2,\dots,n$. We close with some questions concerning the minimal possible number of distinct sums.

Consecutive integers in high-multiplicity sumsets

Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is as follows. Let $k,l\ge 1$ and $n\ge 3$ be integers, and suppose that $A_1,...,A_k\subset[0,l]$ are integer sets of size at least $n$, none of which is contained in an arithmetic progression with difference greater than 1.…

两个平方数之和的连续段

Consecutive runs of sums of two squares

本文研究两个平方数之和序列在算术级数中的连续分布，证明了在可容许同余类条件下，任意长度三段的同余模式以及两段任意长度的同余模式均可出现无穷多次。

A congruence modulo $n^3$ involving two consecutive sums of powers and its applications

For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this note we prove that for each $k\ge 2 $$ 2S_{2k+1}(n)- (2k+1)nS_{2k}(n)\equiv \{{array}{ll} 0\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\,even\,\,or}\,\, n\,\, {\rm is\,\, odd} & {\rm or} \,\, n\equiv 0\,(\bmod{\,4}) \frac{n^3}{2}\,(\bmod{\,n^3}) & {\rm if}\,\,k\,\,{\rm is\,\, odd} &,\,{\rm and}\,\, n\equiv 2\,(\bmod{\,4}). {array}.$$ The above congruence allows us to state an equivalent formulati…

Character analogues of certain Hardy-Berndt sums

In this paper we consider transformation formulas for \[ B\left( z,s:χ\right) =\sum\limits_{m=1}^{\infty}\sum\limits_{n=0} ^{\infty}χ(m)χ(2n+1)\left( 2n+1\right) ^{s-1}e^{πim(2n+1)z/k}. \] We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties of them. With the help of the character analogues of the Euler--Maclaurin summation formula we establish integral representations for the Hardy-Berndt character sums $s_{3,p}\left( d,c:χ\right) $ and $s_{4,p}\left( d,c:χ\right) $.

连续循环和

Consecutive Cycle Sums

本文研究将数字1至n排列在n-循环上，哪些数可表示为连续数字之和，并证明当n≥8时，n-循环不完备。

斐波那契数作为 $k$-广义斐波那契数列中连续项之和

Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence

本文研究 $k$-广义斐波那契数列连续项之和何时为斐波那契数，证明对 $k\geq3$ 此类情形仅有有限多个，并推出两类数列交集有限。

On the Function Field Analogue of Landau's Theorem on Sums of Squares

This paper deals with function field analogues of the famous theorem of Landau which gives the asymptotic density of sums of two squares in $\mathbb{Z}$. We define the analogue of a sum of two squares in $\mathbb{F}_q[T]$ and estimate the number $B_q(n)$ of such polynomials of degree $n$ in two cases. The first case is when $q$ is large and $n$ fixed and the second case is when $n$ is large and $q$ is fixed. Although the methods used and main terms computed in each of the two cases differ, the two iterated limits of (a normalization of) $B_q(n)$ turn out to be exactly the same.

On perfect powers that are sums of cubes of a three term arithmetic progression

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive cubes. Recently, the equation $(x-r)^k + x^k + (x+r)^k$ has been studied for $k=4$ by Zhongfeng Zhang and for $k=2$ by Koutsianas. In this paper, we complement the work of Cassels, Koutsianas and Zhang by considering the case when $k=3$ and showing that the equation $(x-r)^3+x^3+(x+r)^3=y^n$ with $n\geq 5$ a prime and $0 < r \leq 10^6$ only has trivial solu…

GCD of sums of $k$ consecutive squares of generalized Fibonacci numbers

In 2021, Guyer and Mbirika gave two equivalent formulas that computed the greatest common divisor (GCD) of all sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In this current paper, we extend their results to the GCD of all sums of $k$ consecutive squares of these numbers. Denoting these GCD values by the symbol $\mathcal{G}_{G_0, G_1}^2\!(k)$, we prove $\mathcal{G}_{G_0, G_1}^2\!(k) = \gcd\left(G_k G_{k+1} - G_0 G_1,\; G_{k+1}…

GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and l-functions

The aim of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, q-analogue Genocchi type l-function and two-variable q-Genocchi type l-function. Furthermore, we construct new genereting functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations related to q-Hardy-Berndt type sums and q-Genocchi zeta function as well.

On $q$-analogues of two-one formulas for multiple harmonic sums and multiple zeta star values

Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings $(\{2\}^a,c,\{2\}^b)$ that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to $q$-binomial identities and prove $q$-analogues of two-one formulas…

Power values of sums of certain products of consecutive integers and related results

Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$ where $m\in\N_{\geq 2}$ and $a_{1}<a_{2}<\ldots <a_{k}<n$. This equation can be considered as a generalization of the Erdős-Selfridge Diophantine equation $y^m=p_{n}(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\geq 2$ with $a_{1}\geq 2$, then our equation has only finitely many …

关于多重调和 $q$-和的 Maesaka-Seki-Watanabe 型公式的应用

Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic $q$-sums

本文利用 Maesaka-Seki-Watanabe 型公式的 $q$-模拟，给出了多重 zeta 值的 $q$-模拟的对偶性的另一种证明，并证明了 Kawashima 函数的 $q$-模拟的一个恒等式。

寻找连续素数幂和的算法

An Algorithm to Find Sums of Powers of Consecutive Primes

提出一种枚举可表示为连续素数 $k$ 次幂之和的整数 $n\le x$ 的算法，证明其渐近计数界与运行时间界，并给出同量级下界及快速计数算法，扩展了前人关于连续素数平方和的研究。

A p-adic analogue of Siegel's Theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally stud…

通过形式Bernoulli级数推广拉马努金关于连续整数半整数次幂求和的公式

Generalization of Ramanujan's formula for sums of half-integer powers of consecutive integers via formal Bernoulli series

本文通过形式Bernoulli级数推广了Ramanujan关于连续整数半整数次幂求和的公式，扩展了原有结果到所有正半整数次幂。

Transformation formulas of a character analogue of $\logθ_{2}(z)$

In this paper, transformation formulas for the function \[ A_{1}\left(z,s:χ\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}χ\left(n\right)χ\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2πimnz/k} \] are obtained. Sums that appear in transformation formulas are generalizations of the Hardy--Berndt sums $s_{j}(d,c),$ $j=1,2,5$. As applications of these transformation formulas, reciprocity formulas for these sums are derived and several series relations are presented.

Pell和相伴Pell辫序列作为$k$个连续Pell、平衡及相关数之和的最大公约数

Pell and associated Pell braid sequences as GCDs of sums of $k$ consecutive Pell, balancing, and related numbers

研究了六个著名序列（Pell、相伴Pell、平衡及相关数）中$k$个连续项之和的最大公约数，给出了依赖$k$的闭式，并发现其可表示为Pell和相伴Pell辫序列，同时探讨了平方项之和的GCD问题。

费马问题的一个动力系统类比

A Dynamical Analogue of a Question of Fermat

本文研究二次多项式迭代轨道中的连续平方数问题，构造了含三个连续平方数的单参数族，并证明存在x^2+c形式的多项式其轨道含四个连续平方数，同时在标准猜想下给出周期点轨道仅含有理平方数的充要条件。

与除数函数的扭曲和相关的三角模拟恒等式

Trigonometric analogue of the identities associated with twisted sums of divisor functions

本文受 Ramanujan 丢失笔记启发，推导出与除数函数扭曲和相关恒等式的三角函数等价版本，并给出了 $r_6(n)$ 的模拟恒等式。

Fermat quotients: Exponential sums, value set and primitive roots

For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\ge p^{1/2+ε}$ for any fixed $ε>0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$ one can obtain a nontrivial estimate for much shorter sums starting with $N\ge p^ε$. We also obtain lower bounds on …

关于Erdős和Graham的一个关于严格递增序列中连续和的问题

On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences

本文证明了存在常数c>0，使得对任意正整数n，可选取1到n间的递增序列，其连续子段和的不同值至少为cn^2个，回答了Erdős和Graham的问题，并给出上界及解决若干开放问题。

Whipple求和公式的双重和模拟式及多重二项和的受限求和公式

A double-sum analogue of Whipple's summation formula and a restricted sum formula for multiple binomial sums

该研究证明了Whipple的{}_3F_2求和公式的双重和模拟式，并将其应用得到Igarashi参数化多重zeta级数特殊情形下多重二项和的受限求和公式

自然数连续幂和的q-模拟与扩展Carlitz q-Bernoulli数及多项式

$q$-analogues of sums of consecutive powers of natural numbers and extended Carlitz $q$-Bernoulli numbers and polynomials

本文通过构造经典Appell序列的q-模拟框架，扩展了Carlitz q-Bernoulli数与多项式，并利用第二类q-Stirling数给出其显式级数表示和新连接公式。