The Circle Problem in an Arithmetic Progression

1968 ◽  
Vol 11 (2) ◽  
pp. 175-184 ◽  
Author(s):  
R.A. Smith
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Error Term ◽  
Circle Problem ◽  
Image Position

In following a suggestion of S. Chowla to apply a method of C. Hooley [3] to obtain an asymptotic formula for the sum ∑ r(n)r(n+a), where r(n) denotes the number of representations of n≤xn as the sum of two squares and is positive integer, we have had to obtain non-trivial estimates for the error term in the asymptotic expansion of1

XVII.—On the Asymptotic Expansion of the Characteristic Numbers of the Mathieu Equation

1930 ◽  
Vol 49 ◽  
pp. 210-223 ◽  
Author(s):  
Sydney Goldstein
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Mathieu Equation ◽  
Characteristic Number ◽  
Numerical Approximations ◽  
Characteristic Numbers ◽  
The Difference ◽  
Image Position ◽  
Integral Order

An asymptotic formula has recently been given for the characteristic numbers of the Mathieu equation From tabular values, it will be seen that the formula provides good numerical approximations to the characteristic numbers of integral order; but as pointed out by Ince, it provides better approximations to the characteristic numbers of order (m + ½), where m is a positive integer or zero. In this paper we shall first attempt to find out why this should be so, and then go on to show that the formula is probably an asymptotic expansion, in the Poincaré sense, for any characteristic number. A new asymptotic formula is then found for the difference between two characteristic numbers.

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.

scholarly journals A Note on Cube-Full Numbers in Arithmetic Progression

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Mingxuan Zhong ◽  
Yuankui Ma
Keyword(s):  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Error Term ◽  
Exponential Sum

We obtain an asymptotic formula for the cube-full numbers in an arithmetic progression n ≡ l mod   q , where q , l = 1 . By extending the construction derived from Dirichlet’s hyperbola method and relying on Kloosterman-type exponential sum method, we improve the very recent error term with x 118 / 4029 < q .

scholarly journals The error term in the prime orbit theorem for expanding semiflows

2017 ◽  
Vol 38 (5) ◽  
pp. 1954-2000 ◽  
Author(s):  
MASATO TSUJII
Keyword(s):  
Lyapunov Exponent ◽  
Asymptotic Formula ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Error Term ◽  
Maximal Lyapunov Exponent ◽  
Image Position ◽  
Roof Function

We consider suspension semiflows of angle-multiplying maps on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number $\unicode[STIX]{x1D70B}(T)$ of prime periodic orbits with period $\leq T$. The error term is bounded, at least, by $$\begin{eqnarray}\exp \biggl(\biggl(1-\frac{1}{4\lceil \unicode[STIX]{x1D712}_{\text{max}}/h_{\text{top}}\rceil }+\unicode[STIX]{x1D700}\biggr)h_{\text{top}}\cdot T\biggr)\quad \text{in the limit }T\rightarrow \infty\end{eqnarray}$$ for arbitrarily small $\unicode[STIX]{x1D700}>0$, where $h_{\text{top}}$ and $\unicode[STIX]{x1D712}_{\text{max}}$ are, respectively, the topological entropy and the maximal Lyapunov exponent of the semiflow.

scholarly journals Asymptotic expansion of a series of Ramanujan

1992 ◽  
Vol 35 (2) ◽  
pp. 189-199
Author(s):  
Bruce C. Berndt ◽  
Ronald J. Evans
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Image Position

An asymptotic expansion is given for the seriesas x→∞ in the sector |Argx|≦π/2–δ. Here δ, Re(a), and Re(s) are positive and r is a positive integer. In the case a = r = s = 1, this yields the nontrivial resultstated by Ramanujan in his notebooks [6].

scholarly journals On a generalisation of a result of Ramanujan connected with the exponential series

1981 ◽  
Vol 24 (3) ◽  
pp. 179-195
Author(s):  
R. B. Paris
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Exponential Series ◽  
Large Positive Integer ◽  
The Many ◽  
Image Position

One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined byRamanujan (9) showed that when n is large, θn possesses the asymptotic expansionThe first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined bythen πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansionA generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by

scholarly journals Squarefree Integers in Arithmetic Progressions to Smooth Moduli

2021 ◽  
Vol 9 ◽  
Author(s):  
Alexander P. Mangerel
Keyword(s):  
Asymptotic Formula ◽  
Arithmetic Progression ◽  
Error Term ◽  
Prime Power ◽  
Exponential Sums ◽  
Power Saving ◽  
Positive Density ◽  
Squarefree Integer ◽  
Squarefree Integers ◽  
Prime Case

Abstract Let $\varepsilon> 0$ be sufficiently small and let $0 < \eta < 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

scholarly journals A note on the 2 k-th mean value of the Hurwitz zeta function

1999 ◽  
Vol 60 (3) ◽  
pp. 403-405 ◽  
Author(s):  
A. Kumchev
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Error Term ◽  
Mean Value ◽  
Substantial Improvement ◽  
Hurwitz Zeta Function ◽  
Image Position

Consider the error term in the asymptotic formulaIn this note we obtain δ(k) ≍ 1/(k6 log2k) which, for large values of k, presents a substantial improvement over the previously known result .

scholarly journals The least common multiple of consecutive arithmetic progression terms

2011 ◽  
Vol 54 (2) ◽  
pp. 431-441 ◽  
Author(s):  
Shaofang Hong ◽  
Guoyou Qian
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

Asymptotic formulae for solutions of linear second-order differential equations with a large parameter

1960 ◽  
Vol 1 (4) ◽  
pp. 439-464 ◽  
Author(s):  
R. C. Thorne
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Positive Integer ◽  
Complex Variable ◽  
Regular Function ◽  
Arbitrary Parameter ◽  
Large Parameter ◽  
Second Order Differential Equations ◽  
Asymptotic Formulae ◽  
Image Position

AbstractUniform asymptotic formulae are obtained for solutions of the differential equation for large positive values of the parameter u. Here p is a positive integer, θ an arbitrary parameter and z a complex variable whose domain of variation may be unbounded. The function ƒ (u, θ, z) is a regular function of ζ having an asymptotic expansion of the form for large u.The results obtained include and extend those of earlier writers which are applicable to this equation.

Sign in / Sign up

Export Citation Format

Share Document