Abstract: According to the Erdős discrepancy conjecture, for any infinite ±1 sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any ±1 sequence (x 1,x 2,...) and a discrepancy C, there exist integers m and d such that \(|\sum_{i=1}^m x_{i \cdot d}| > C\). This is an 80-year-old open problem and recent development proved that this conjecture is true for discrepancies up to 2. Paul Erdős also conjectured that this property of unbounded discrepancy even holds for the restricted case of completely multiplicative sequences, namely sequences (x 1,x 2,...) where x a ·b = x a ·x b for any a,b ≥ 1. The longest such sequence of discrepancy 2 has been proven to be of size 246. In this paper, we prove that any completely multiplicative sequence of size 127,646 or more has discrepancy at least 4, proving the Erdős discrepancy conjecture for discrepancy up to 3. In addition, we prove that this bound is tight and increases the size of the longest known sequence of discrepancy 3 from 17,000 to 127,645. Finally, we provide inductive construction rules as well as streamlining methods to improve the lower bounds for sequences of higher discrepancies.

Abstract: We introduce a new nonparametric outlier detection method for linear series, which requires no missing or removed data imputation. For an arithmetic progression (a series without outliers) with n elements, the ratio (R) of the sum of the minimum and the maximum elements and the sum of all elements is always 2/n : (0,1]. R ≠ 2/n always implies the existence of outliers. Usually, R 2/n implies that the maximum is an outlier. Based upon this, we derived a new method for identifying significant and nonsignificant outliers, separately. Two different techniques were used to manage missing data and removed outliers: (1) recalculate the terms after (or before) the removed or missing element while maintaining the initial angle in relation to a certain point or (2) transform data into a constant value, which is not affected by missing or removed elements. With a reference element, which was not an outlier, the method detected all outliers from data sets with 6 to 1000 elements containing 50% outliers which deviated by a factor of ±1.0e - 2 to ±1.0e + 2 from the correct value.

Abstract: In this paper, we study free probability on the algebra \(\mathcal{A }\) consisting of all arithmetic functions, determined by the gaps between primes. As a continuation and as an application of Cho (Classification on Arithmetic Functions and Free-Moment \(L\) -Functions. Submitted to B. Korean Math Soc, 2013), we study moment series and R-transforms induced by both arithmetic functions and gaps between primes. Also, we consider R-transform calculus on the arithmetic algebra.

Abstract: We prove that if A ⊆ {1, ..., N} has no nontrivial solution to the equation x 1 + x 2 + x 3 + x 4 + x 5 = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \), c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.

Abstract: By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.

Abstract: In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $aw([n],k)$ denotes the smallest number of colors with which the integers $\{1,\ldots,n\}$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. We establish that $aw([n],3)=\Theta(\log n)$ and $aw([n],k)=n^{1-o(1)}$ for $k\geq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can "wrap around," and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungi\'c et al., \textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $k\geq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.

Abstract: For relatively prime positive integers and , we consider the least common multiple of the finite arithmetic progression . We derive new lower bounds on that improve upon those obtained previously when either or is large. When is prime, our best bound is sharp up to a factor of for properly chosen, and is also nearly sharp as .

Abstract: In this paper, we derive various formulae for the sum of k-Jacobsthal numbers with indexes in an arithmetic sequence, say an+r for fixed integers a and r Also, we describe generating function and the alternated sum formula for k-Jacobsthal numbers with indexes in an arithmetic sequence.

Abstract: Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1/6+\epsilon}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least one prime $p\in(y,y+h]$ lying in every reduced arithmetic progression $a\mod q$, provided that $1\le Q^2\le h/x^{1/15+\epsilon}$.

Abstract: Digital roots of numbers have several interesting properties, most of which are well-known In this paper, our goal is to prove some lesser known results concerning the digital roots of powers of numbers in an arithmetic progression We will also state some theorems concerning the digital roots of Fermat numbers and star numbers We will conclude our paper by an interesting application

Abstract: We investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound, and lying in some arithmetic progression a (mod q). In particular, we prove an upper bound for its variance as a varies over $(\mathbb{Z}/q\mathbb{Z})^{\times}$ which considerably improves upon earlier work of Blomer.

Abstract: Let be the number of squares in the arithmetic progression , for , , and let be the maximum of over all non-trivial arithmetic progressions . Rudin’s conjecture claims that , and in its stronger form that if . We prove the conjecture above for . We even prove that the arithmetic progression is the only one, up to equivalence, that contains squares for the values of such that increases, for ( and ). Supplementary materials are available with this article.

Abstract: An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms $a_0,a_1,\ldots,a_k$ and with each subsequent term $a_n>a_{n-1}$ chosen minimally such that the 3-free condition is not violated. Odlyzko and Stanley conjectured that Stanley sequences divide into two classes based on asymptotic growth patterns, with one class of highly structured sequences satisfying $a_n\approx \Theta(n^{\log_2 3})$ and another class of seemingly chaotic sequences obeying $a_n=\Theta(n^2/\log n)$. We propose a rigorous definition of regularity in Stanley sequences based on local structure rather than asymptotic behavior and show that our definition implies the corresponding asymptotic property proposed by Odlyzko and Stanley. We then construct many classes of regular Stanley sequences, which include as special cases all such sequences previously identified. We show how two regular sequences may be combined into another regular sequence, and how parts of a Stanley sequence may be translated while preserving regularity. Finally, we demonstrate that certain Stanley sequences possess proper subsets that are also Stanley sequences, a situation that appears previously to have been assumed impossible.

Abstract: In this note we are interested in the problem of whether or not every increasing sequence of positive integers x1x2x3 with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms xi, x j, and xk such that i+ k = 2 j and xi+ xk = 2x j. We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present several results obtained by using RamseyScript, a high-level scripting language.

Abstract: This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature.

Abstract: In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the average value of $A(n-h)B(n)$ over any arithmetic progression is only dependent on the common difference of the progression. We use this method on the problem of finding mean value of $K(N)$, where $K(N)/\log N$ is the expected number of primes such that a random elliptic curve over rationals has $N$ points when reduced over those primes.

Abstract: Fibonacci numbers are fascinating and their impact on the field of mathematics has been great. In this paper, mainly present formulas for the sums of k-Lucas numbers with indexes in an arithmetic sequence, say an+r, for fixed integers a and r (0 \leq r \leq a-1). Also the generating function evaluates and presents the alternating sum for the k-Lucas numbers with arithmetic index.

Abstract: In this paper, the authors has proved the solution of a new type of functional equation f ( k ∑ j=1 jp xj ) = k ∑ j=1 ( jp f (xj) ) , k, p ≥ 1 which is originating from sum of higher powers of an arithmetic progression. Its generalized Ulam Hyers stability in Banach space using direct and fixed point methods are investigated. An application of this functional equation is also studied.

Abstract: Recently Holliday and Komatsu extended the results of Ohtsuka and Nakamura on reciprocal sums of Fibonacci numbers to reciprocal sums of generalized Fibonacci numbers. The aim of this work is to give similar results for the alternating sums of reciprocals of the generalized Fibonacci numbers with indices in arithmetic progression. Finally we note our generalizations of some results of Holliday and Komatsu.

Abstract: This is the first of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.

Abstract: Expressions for finite sums involving the binomial coefficients with unit fraction coefficients whose denominators form an arithmetic sequence are determined.

Abstract: This research paper is a general overview of Sequence and Series. In this, we have studied about Progression s and their types which include Arithmetic Progression, Geometric Progression and Harmonic Progression.

Abstract: Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval $(y,y+h]$, provided that $h>x^{1/6+\epsilon}$ and $Q$ satisfies appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least one prime $p\in(y,y+h]$ lying in every reduced arithmetic progression $a\mod q$, provided that $1\le Q^2\le h/x^{1/15+\epsilon}$.

Abstract: Let B be a set of real numbers of size n . We prove that the length of the longest arithmetic progression contained in the product set B.B={bibj|bi,bj∈B}B.B={bibj|bi,bj∈B} cannot be greater than View the MathML sourceO(n1+1/loglogn) an arithmetic progression of length View the MathML sourceΩ(nlogn), so the obtained upper bound is close to the optimal.

Abstract: The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of the primes such as primes of the form $\lfloor \pi n\rfloor$ and some of arithmetic density zero such as primes of the form $\lfloor n\log\log n\rfloor$.

Abstract: We present a method for optimization with sums of exponentials subject to positivity constraints and apply it to the modeling of empirical probability distribution functions and to the design of IIR filters with non-negative impulse response. Our approach uses exponents in a sparse arithmetic progression and hence is able to transform the positivity condition to a polynomial form that is computationally tractable. We show how to obtain initial values for the exponents by sparsifying a full progression and then present an iterative optimization procedure using gradient steps. The modeling and design examples indicate a good behavior of our method.

Abstract: We take a well-known problem—which numbers can or cannot be written as a sum of consecutive integers—and generalize it for summations involving any arithmetic sequence. Various general and specific...