scholarly journals COUNTING -FIELDS WITH A POWER SAVING ERROR TERM

2014 ◽  
Vol 2 ◽  
Author(s):  
ARUL SHANKAR ◽  
JACOB TSIMERMAN
Keyword(s):  
Wide Class ◽  
Error Term ◽  
Counting Function ◽  
Power Saving ◽  
Geometry Of Numbers ◽  
Counting Problems ◽  
Error Terms

AbstractWe show how the Selberg $\Lambda ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using the geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.

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.

Error terms for closed orbits of hyperbolic flows

2001 ◽  
Vol 21 (2) ◽  
pp. 545-562 ◽  
Author(s):  
MARK POLLICOTT ◽  
RICHARD SHARP
Keyword(s):  
Error Term ◽  
Closed Orbit ◽  
Counting Function ◽  
Diophantine Condition ◽  
Weak Mixing ◽  
Closed Orbits ◽  
Hyperbolic Flows ◽  
Error Terms ◽  
Anosov Flows

In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

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 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 A simple solution of the spurious regression problem

2017 ◽  
Vol 22 (3) ◽  
Author(s):  
Cindy Shin-Huei Wang ◽  
Christian M. Hafner
Keyword(s):  
Wide Class ◽  
General Framework ◽  
Error Term ◽  
Applied Economics ◽  
Exact Form ◽  
Two Stage ◽  
Regression Problem ◽  
Spurious Regression ◽  
Long Run ◽  
Simulation Results

Abstract This paper develops a new estimator for cointegrating and spurious regressions by applying a two-stage generalized Cochrane-Orcutt transformation based on an autoregressive approximation framework, even though the exact form of the error term is unknown in practice. We prove that our estimator is consistent for a wide class of regressions. We further show that a convergent usual t-statistic based on our new estimator can be constructed for the spurious regression cases analyzed by (Granger, C. W. J., and P. Newbold. 1974. “Spurious Regressions in Econometrics.” Journal of Econometrics 74: 111–120) and (Granger, C. W. J., N. Hyung, and H. Jeon. 2001. “Spurious Regressions with Stationary Series.” Applied Economics 33: 899–904). The implementation of our estimator is easy since it does not necessitate estimation of the long-run variance. Simulation results indicate the good statistical properties of the new estimator in small and medium samples, and also consider a more general framework including multiple regressors and endogeneity.

scholarly journals On the Hooley’s problem on the representation of a number as the sum of a square and a product

2019 ◽  
Vol 485 (5) ◽  
pp. 539-544
Author(s):  
V. A. Bykovskii ◽  
A. V. Ustinov
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Power Saving ◽  
Number Of Solutions ◽  
First Time

The article is devoted to the Hooley’s problem on the representation of a number as the sum of a square and a product. For the first time we show that number of solutions satisfy an asymptotic formula with power saving in error term.

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.

Multiplicative Models For Continuous Dependent Variables: Estimation on Unlogged versus Logged Form

2017 ◽  
Vol 47 (1) ◽  
pp. 113-164 ◽  
Author(s):  
Trond Petersen
Keyword(s):  
Error Term ◽  
Geometric Mean ◽  
Sufficient Conditions ◽  
Arithmetic Mean ◽  
Sociological Research ◽  
Sufficient Condition ◽  
Statistical Independence ◽  
Unbiased Estimators ◽  
Independent Variables ◽  
Error Terms

In regression analysis with a continuous and positive dependent variable, a multiplicative relationship between the unlogged dependent variable and the independent variables is often specified. It can then be estimated on its unlogged or logged form. The two procedures may yield major differences in estimates, even opposite signs. The reason is that estimation on the unlogged form yields coefficients for the relative arithmetic mean of the unlogged dependent variable, whereas estimation on the logged form gives coefficients for the relative geometric mean for the unlogged dependent variable (or for absolute differences in the arithmetic mean of the logged dependent variable). Estimated coefficients from the two forms may therefore vary widely, because of their different foci, relative arithmetic versus relative geometric means. The first goal of this article is to explain why major divergencies in coefficients can occur. Although well understood in the statistical literature, this is not widely understood in sociological research, and it is hence of significant practical interest. The second goal is to derive conditions under which divergencies will not occur, where estimation on the logged form will give unbiased estimators for relative arithmetic means. First, it derives the necessary and sufficient conditions for when estimation on the logged form will give unbiased estimators for the parameters for the relative arithmetic mean. This requires not only that there is arithmetic mean independence of the unlogged error term but that there is also geometric mean independence. Second, it shows that statistical independence of the error terms on regressors implies that there is both arithmetic and geometric mean independence for the error terms, and it is hence a sufficient condition for absence of bias. Third, it shows that although statistical independence is a sufficient condition, it is not a necessary one for lack of bias. Fourth, it demonstrates that homoskedasticity of error terms is neither a necessary nor a sufficient condition for absence of bias. Fifth, it shows that in the semi-logarithmic specification, for a logged error term with the same qualitative distributional shape at each value of independent variables (e.g., normal), arithmetic mean independence, but heteroskedasticity, estimation on the logged form will give biased estimators for the parameters for the arithmetic mean (whereas with homoskedasticity, and for this case thus statistical independence, estimators are unbiased, from the second result above).

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