scholarly journals An Improved Asymptotic on the Representations of Integers as Sums of Products

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Wenjia Zhao
Keyword(s):  
Upper Bound ◽  
Error Term ◽  
Error Terms ◽  
Representations Of Integers ◽  
Sums Of Products

In this paper, we improve the error terms of Chace’s results in the study by Chace (1994) on the number of ways of writing an integer N as a sum of k products of l factors, valid for k ≥ 3 and l = 2 , 3. More precisely, for l = 2 , 3, we improve the upper bound N k − 1 − 2 k − 2 / k − 1 l + 1 + ε , k ≥ 3 for the error term, to N 2 − 2 / 2 l + 1 + ε when k = 3 and N k − 1 − 4 k − 2 / l + 1 k + l − 2 + ε when k ≥ 4 .

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

scholarly journals Densities in certain three-way prime number races

2020 ◽  
pp. 1-34
Author(s):  
Jiawei Lin ◽  
Greg Martin
Keyword(s):  
Asymptotic Formula ◽  
Prime Number ◽  
Error Term ◽  
Proof Technique ◽  
Real Numbers ◽  
Logarithmic Density ◽  
Mutual Independence ◽  
Error Terms ◽  
The Relationship

Abstract Let $a_1$ , $a_2$ , and $a_3$ be distinct reduced residues modulo q satisfying the congruences $a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$ . We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which $\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$ . The relationship among the $a_i$ allows us to normalize the error terms for the $\pi (x;q,a_i)$ in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.

scholarly journals Estimates of convolutions of certain number-theoretic error terms

2004 ◽  
Vol 2004 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Aleksandar Ivic
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Abelian Groups ◽  
Divisor Problem ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Convolution Function

Several estimates for the convolution functionC [f(x)]:=∫1xf(y) f(x/y)(dy/y)and its iterates are obtained whenf(x)is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for∫0T|ζ(1/2+it)|2kdt(k=1,2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin-Selberg convolution.

scholarly journals An $\Omega$-result related to $r_4(n)$.

1989 ◽  
Vol Volume 12 ◽  
Author(s):  
Sukumar Das Adhikari ◽  
R Balasubramanian ◽  
A Sankaranarayanan
Keyword(s):  
Upper Bound ◽  
Error Term ◽  
Elementary Proof ◽  
Arithmetical Function ◽  
Average Order ◽  
Elementary Method ◽  
International Audience

International audience Let $r_4(n)$ be the number of ways of writing $n$ as the sum of four squares. Set $P_4(x)= \sum \limits_{n\le x} r_4(n)-\frac {1}{2}\pi^2 x^2$, the error term for the average order of this arithmetical function. In this paper, following the ideas of Erd\"os and Shapiro, a new elementary method is developed which yields the slightly stronger result $P_4(x)= \Omega_{+}(x \log \log x)$. We also apply our method to give an upper bound for a quantity involving the Euler $\varphi$-function. This second result gives an elementary proof of a theorem of H. L. Montgomery

scholarly journals Regression Analysis As Analytical Procedures: A Simulation Study Of The Effect Of Error Term Nonnormality On Auditor Decisions

2011 ◽  
Vol 7 (2) ◽  
pp. 51
Author(s):  
Arlette C. Wilson
Keyword(s):  
Regression Analysis ◽  
Error Term ◽  
Decision Rules ◽  
Simulated Data ◽  
Audit Tool ◽  
Underlying Assumption ◽  
Audit Period ◽  
Error Terms ◽  
Period Data ◽  
Regression Theory

Regression analysis has been shown through research to be a useful audit tool. One underlying assumption of regression theory is error-term normality. Decision rules were applied to models developed using simulated data with normal and nonnormal error terms. Material errors were seeded into the audit period data, and incorrect rejections and acceptances were tabulated for both types of models. The results of this study suggest that nonnormality of error terms may not significantly affect auditor decisions.

scholarly journals SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

2018 ◽  
Vol 6 ◽  
Author(s):  
THOMAS A. HULSE ◽  
CHAN IEONG KUAN ◽  
DAVID LOWRY-DUDA ◽  
ALEXANDER WALKER
Keyword(s):  
Error Term ◽  
Power Saving ◽  
Lattice Points ◽  
Dimensional Sphere ◽  
Mean Square ◽  
Circle Problem ◽  
Second Moments ◽  
The Laplace Transform ◽  
Error Terms ◽  
Gauss Circle Problem

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to$P_{k}(n)^{2}$, where$P_{k}(n)$is the discrepancy between the volume of the$k$-dimensional sphere of radius$\sqrt{n}$and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including$\sum P_{k}(n)^{2}e^{-n/X}$and the Laplace transform$\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions$k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums$\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral$\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

ON THE ERROR TERM IN THE APPROXIMATE FUNCTIONAL EQUATION FOR EXPONENTIAL SUMS RELATED TO CUSP FORMS

2008 ◽  
Vol 04 (05) ◽  
pp. 747-756 ◽  
Author(s):  
ANNE-MARIA ERNVALL-HYTÖNEN
Keyword(s):  
Functional Equation ◽  
Upper Bound ◽  
Error Term ◽  
Exponential Sums ◽  
Cusp Forms ◽  
Approximate Functional Equation ◽  
Holomorphic Cusp Forms

We give a proof for the approximate functional equation for exponential sums related to holomorphic cusp forms and derive an upper bound for the error term.

On Riesz means of the coefficients of the Rankin–Selberg series

1999 ◽  
Vol 127 (1) ◽  
pp. 117-131 ◽  
Author(s):  
ALEKSANDAR IVIĆ ◽  
KOHJI MATSUMOTO ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Error Term ◽  
Weighted Sum ◽  
Mean Square ◽  
Riesz Means ◽  
The Mean ◽  
Voronoi Formula

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

scholarly journals Convolutions and mean square estimates of certain number-theoretic error terms

2006 ◽  
Vol 80 (94) ◽  
pp. 141-156 ◽  
Author(s):  
Aleksandar Ivic
Keyword(s):  
Number Theory ◽  
Error Term ◽  
Analytic Number Theory ◽  
Upper Bounds ◽  
Mean Square ◽  
Analytic Number ◽  
Error Terms ◽  
Convolution Function

We study the convolution function C[f(x)]:=\int_1^x f(y)f\Bigl(\frac xy\Bigr)\frac{dy}y When f(x) is a suitable number-theoretic error term. Asymptotics and upper bounds for C[f(x)] are derived from mean square bounds for f(x). Some applications are given, in particular to |\zeta(\tfrac12+ix)|^{2k} and the classical Rankin--Selberg problem from analytic number theory.

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