In this note, we first review the novel approach to power sums put forward recently by Muschielok in arXiv:2207.01935v1, which can be summarized by the formula $S_m^{(a)}(n) = \sum_{k} c_{mk} ψ_k^{(a)}(n)$, where the $c_{mk}$'s are the expansion coefficients and where the basis functions $ψ_m^{(a)}(n)$ fulfil the recursive property $ψ_m^{(a+1)}(n)= \sum_{i=1}^n ψ_m^{(a)}(i)$. Then, we point out a number of supplementary facts concerning the said approach not contemplated explicitly in Muschielok's paper. In particular, we show that, for any given $m$, the values of the $c_{mk}$'s can be obtained by inverting a matrix involving only binomial coefficients. This may be compared with the original approach of Muschielok, where the values of the $c_{mk}$'s can be obtained by inverting a lower triangular matrix involving the Stirling numbers of the first kind. Also, we make a conjecture about the functional form of the coefficients $c_{m\, m-k}$.
it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.
In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients.
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.
Kannan Soundararajan recently obtained a new estimate, conditional to the Riemann hypothesis, for the summatory function of the Mobius function. In this expository article we describe his method, with detailed computations.
In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{ϕ([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.
The Ohno relation is a well-known relation among multiple zeta values. Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relation and presented two types of new relations and five conjectural formulas. This paper proves one of these formulas.
Following the work of Hildebrand we improve the Po'lya- Vinogradov inequality in a specific range, we also give a general result that shows its dependency on Burgess bound and at last we improve the range of validity for a special case of Burgess' character sum estimate.
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.
Recently, the explicit evaluation of Gauss sums in the index 2 and 4 cases have been given in several papers (see [2,3,7,8]). In the course of evaluation, the sigh (or unit root) ambiguities are unavoidably occurred. This paper presents another method, different from [7] and [8], to determine the sigh (unit root) ambiguities of Gauss sums in the index 2 case, as well as the ones with odd order in the non-cyclic index 4 case. And we note that the method in this paper are more succinct and effective than [8] and [7].
In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight alpha by using p-adic invariant q-integral on Zp.
In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that $\max_{ θ\in [0,1) }\left|\sum_{ n \leq x } μ(n) e(n θ) \right| \ll_εx^{3/4 + ε}$. The purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.
For a given positive integers $m$ and $\ell$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2 \leq n \leq m$, there are distinct $m$th roots of unity $x_1,\dots,x_n \in \mathbb{C}$ such that $x_1^{\ell} + \cdots + x_n^{\ell}=0$.
We prove an upper bound for the exponential sum associated to a localized $k-$divisor function, i.e., the counting function of the number of ways to write a positive integer $n$ as a product of $k\ge 2$ positive integers, each of them belonging to a specified interval. In particular, this gives an estimate for the exponential sum for the $k-$divisor function, $d_k(n)$.
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.
In this note, we deduce an asymptotic formula for even power moments of Kloosterman sums based on the important work of N. M. Katz on Kloosterman sheaves. In a similar manner, we can also obtain an upper bound for odd power moments. Moreover, we shall give an asymptotic formula for odd power moments of absolute Kloosterman sums. Consequently, we find that there are infinitely many $a\bmod p$ such that $S(a,1;p)\gtrless0$ as $p\rightarrow+\infty.$
Ramanujan sums have been studied and generalized by several authors. For example, Nowak studied these sums over quadratic number fields, and Grytczuk defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.
We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.
We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.
In this paper, we will give another proof of Zhi-Wei Sun's three conjectures on Apéry-like sums involving harmonic numbers by proving some identities among special values of multiple polylogarithms.
Let $A\subset\mathbb{N}$, $α\in(0,1)$, and for $x\in\mathbb{R}$ let $e(x):=e^{2πix}$. We set $$S_{A}(α,N):=\sum_{\substack{n\in A\n\leq N}}e(nα).$$ Recently, Lambert A'Campo proposed the following question: is there an infinite non-cofinite set $A\subset\mathbb{N}$ such that for all $α\in(0,1)$ the sum $S_{A}(α,N)$ has bounded modulus as $N\to +\infty$? In this note we show that such sets do not exist. To do so, we use a theorem by Duffin and Schaeffer on complex power series. We extend our result by proving that if the sum $S_{A}(α,N)$ is bounded in modulus on an arbitrarily small interval an…
In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl's bound for exponential sums and a device used by Popov.
A prime number $p$ is called a Schenker prime if there exists such $n\in\mathbb{N}_+$ that $p\nmid n$ and $p\mid a_n$, where $a_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j$ is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ in case when $p$ is a Schenker prime. In particular, they asked whether for each $k\in\mathbb{N}_+$ there exists the unique positive integer $n_k<p^k$ such that $v_p(a_{m\cdot 5^k + n_k})\geq k$ for each nonnegative integer $m$. We prove that for every $k\in\mathbb{N}_+$ the inequality $v_5(a_n)\geq k$ h…
In this paper we introduce the notion of the $P$-sequences and apply their properties in studying representability of real numbers. Another application of $P$-sequences we find in generating the Prouhet-Tarry-Escott pairs.
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.
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 $\geq 2$.
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly a…
This note simplifies the proof of a recent result on the oscillation of the prime product in Martens Theorem, and provides a quantitative expression for the error term. In addition, the corresponding oscillation results for the finite sums of the reciprocal of the primes up to a fixed number and other finite sums and products are also given.
In this note, we characterize all functions $f : \mathbb{N} \rightarrow \mathbb{C}$ such that $f(x_1^2+ \cdots + x_k^2)=f(x_1)^2+ \cdots + f(x_k)^2$, where $k \geq 3$ and $x_1, \cdots, x_k$ are positive integers.
It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.
In a recent note W. Kohnen asks whether the values of Dedekind sums are dense in the field of $p$-adic numbers. The present paper answers this question. Dedekind sums do not approximate units of $\mathbb Z_2$ or $\mathbb Z_3$, so they are not dense in $\mathbb Q_2$ or $\mathbb Q_3$. But they are dense in $\mathbb Q_p$ if $p\ge 5$.
In this note we consider the theorem established in arXiv:1912.07171 concerning the sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and show that it can be used to demonstrate the classical theorem of Faulhaber for both cases of odd and even $k$.
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.
In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the integer and the adjacent integer(s) has a constant sum $x$ (respectively $y$) with $x\ne y$.
In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretèche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.
We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} χ(n)$, where $χ$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty$ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $χ$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true…
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.
Multiple zeta values (MZVs) with certain repeated arguments or certain sums of cyclically generated MZVs are evaluated as rational multiple of powers of $π^2$. In this paper, we give a short and simple proof of the remarkable evaluations of MZVs established by D. Borman and D. M. Bradley.
We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.
Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.
