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.

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.

scholarly journals The Laplace transform of the second moment in the Gauss circle problem

2021 ◽  
Vol 15 (1) ◽  
pp. 1-27
Author(s):  
Thomas A. Hulse ◽  
Chan Ieong Kuan ◽  
David Lowry-Duda ◽  
Alexander Walker
Keyword(s):  
Laplace Transform ◽  
Second Moment ◽  
Circle Problem ◽  
The Laplace Transform ◽  
Gauss Circle Problem

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 Optimal spectral rectangles and lattice ellipses

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120492 ◽  
Author(s):  
Pedro R. S. Antunes ◽  
Pedro Freitas
Keyword(s):  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Error Term ◽  
Dirichlet Boundary ◽  
Unit Area ◽  
Integer Lattice ◽  
Lattice Points ◽  
Computational Method ◽  
Dirichlet Boundary Conditions ◽  
Circle Problem

We consider the problem of minimizing the k th eigenvalue of rectangles with unit area and Dirichlet boundary conditions. This problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant. We show that, as k goes to infinity, the optimal rectangle approaches the square and, correspondingly, the optimal ellipse approaches the circle. We also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for Gauss's circle problem.

scholarly journals On the mean square formula of the error term in the Dirichlet divisor problem

2009 ◽  
Vol 146 (2) ◽  
pp. 277-287 ◽  
Author(s):  
YUK-KAM LAU ◽  
KAI-MAN TSANG
Keyword(s):  
Remainder Term ◽  
Error Term ◽  
Divisor Problem ◽  
Mean Square ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Riemann Zeta ◽  
The Mean ◽  
Upper Estimate

AbstractLet F(x) be the remainder term in the mean square formula of the error term Δ(t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function.

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 GENERALIZATION OF THE EFFECTIVE WIENER–IKEHARA THEOREM

2013 ◽  
Vol 09 (08) ◽  
pp. 2091-2128 ◽  
Author(s):  
SZILÁRD GY. RÉVÉSZ ◽  
ANNE de ROTON
Keyword(s):  
Laplace Transform ◽  
Tauberian Theorem ◽  
Error Term ◽  
Effective Version ◽  
Slow Decrease ◽  
Asymptotic Evaluation ◽  
The Laplace Transform

We consider the classical Wiener–Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on the boundary of convergence of the Laplace transform. In this generality, we prove the otherwise known asymptotic evaluation of the transformed function, when the usual conditions of the Wiener–Ikehara theorem hold. However, our version also provides an effective error term, not known thus far in this generality. The crux of the proof is a proper, asymptotic variation of the lemmas of Ganelius and Tenenbaum, also constructed for the sake of an effective version of the Wiener–Ikehara theorem.

ON A NEW CIRCLE PROBLEM

2016 ◽  
Vol 103 (2) ◽  
pp. 231-249
Author(s):  
JUN FURUYA ◽  
MAKOTO MINAMIDE ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Dirichlet Character ◽  
Arithmetical Function ◽  
Circle Problem ◽  
The Riemann Zeta Function ◽  
Primitive Dirichlet Character ◽  
Riemann Zeta ◽  
The Mean ◽  
Voronoi Formula

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

Distribution of the error term for the number of lattice points inside a shifted ball

1996 ◽  
pp. 141-153 ◽  
Author(s):  
P. Bleher ◽  
J. Bourgain
Keyword(s):  
Error Term ◽  
Lattice Points

A twist of the Gauss circle problem by holomorphic cusp forms

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ratnadeep Acharya
Keyword(s):  
Cusp Forms ◽  
Circle Problem ◽  
Holomorphic Cusp Forms ◽  
Gauss Circle Problem
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