On the restricted divisor function in arithmetic progressions

Igor E. Shparlinski

Type: Article

Publication Date: 2012-01-20

Citations: 6

DOI: https://doi.org/10.4171/rmi/675

Abstract

We obtain several asymptotic estimates for the sums of the restricted divisor function \tau_{M,N}(k) = \# \{1 \le m \le M, \ 1\le n \le N : mn = k\} over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function \alpha k^2 , k=1,2,\dots with a real \alpha .

Locations

Revista Matemática Iberoamericana - View - PDF
arXiv (Cornell University) - View - PDF

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Works Cited by This (14)

Action	Title	Year	Authors
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+	The congruence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mspace width="0.25em"/><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">mod</mml:mi><mml:mspace width="0.25em"/><mml:mi>m</mml…	2009	Todd Cochrane Sanying Shi
+	The equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>λ</mml:mi></mml:math> in fields of prime order and applications	2008	M. Z. Garaev V. C. Garcia
+	THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS	2007	Valentin Blomer
+ PDF Chat	Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average	2008	Igor E. Shparlinski
+	Incomplete Kloosterman Sums and a Divisor Problem	1985	John Friedlander Henryk Iwaniec
+ PDF Chat	The distribution of spacings between the fractional parts of n 2 α	2001	Zeév Rudnick Peter Sarnak Alexandru Zaharescu
+ PDF Chat	Divisor Problems and the Pair Correlation for the Fractional Parts of n2	2010	Jimi Lee Truelsen
+ PDF Chat	Pair correlation for fractional parts of α<i>n</i><sup>2</sup>	2010	D. R. Heath‐Brown
+	An Introduction to the Theory of Numbers. By G. H. Hardy and E. M. Wright. 2nd edition. Pp. xvi, 407 25s. 1945. (Oxford)	1946	T. A. A. B.