Explicit bound of the divisor function

2020 ◽  
Vol 63 (1) ◽  
pp. 133-136
Author(s):  
Thomas Dubbe
Keyword(s):  
Divisor Function ◽  
Explicit Bound

scholarly journals IMPROVING AN INEQUALITY FOR THE DIVISOR FUNCTION

2018 ◽  
Vol 97 (3) ◽  
pp. 374-381
Author(s):  
JEFFREY P. S. LAY
Keyword(s):  
American Mathematical Society ◽  
Mathematical Society ◽  
Divisor Function ◽  
Explicit Bound

Using elementary means, we improve an explicit bound on the divisor function due to Friedlander and Iwaniec [Opera de Cribro, American Mathematical Society, Providence, RI, 2010]. Consequently, we modestly improve a result regarding a sieving inequality for Gaussian sequences.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

scholarly journals n-Color partitions with weighted differences equal to minus two

1997 ◽  
Vol 20 (4) ◽  
pp. 759-768 ◽  
Author(s):  
A. K. Agarwal ◽  
R. Balasubrananian
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Combinatorial Identities ◽  
Combinatorial Identity ◽  
Divisor Function ◽  
Open Problems

In this paper we study thosen-color partitions of Agarwal and Andrews, 1987, in which each pair of parts has weighted difference equal to−2Results obtained in this paper for these partitions include several combinatorial identities, recurrence relations, generating functions, relationships with the divisor function and computer produced tables. By using these partitions an explicit expression for the sum of the divisors of odd integers is given. It is shown how these partitions arise in the study of conjugate and self-conjugaten-color partitions. A combinatorial identity for self-conjugaten-color partitions is also obtained. We conclude by posing several open problems in the last section.

scholarly journals Note on divisor function for quaternion algebras

2010 ◽  
Vol 130 (10) ◽  
pp. 2147-2156
Author(s):  
Guangshi Lü ◽  
Honggang Xia
Keyword(s):  
Divisor Function ◽  
Quaternion Algebras

scholarly journals A Combinatorial Proof of an Identity for the Divisor Generating Function

10.37236/2153 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Masanori Ando
Keyword(s):  
Generating Function ◽  
Combinatorial Proof ◽  
Divisor Function

In this paper, we give combinatorial proofs and new generalizations of $q$-series identities of Dilcher and Uchimura related to divisor function. Some interesting combinatorial results related to partition and arm-length are also presented.

scholarly journals Omega theorems for the twisted divisor function

2020 ◽  
Vol 62 (2) ◽  
pp. 171-186
Author(s):  
Kamalakshya Mahatab ◽  
Anirban Mukhopadhyay
Keyword(s):  
Divisor Function

scholarly journals An Elementary Proof of Jin's Theorem with a Bound

10.37236/3846 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Mauro Di Nasso
Keyword(s):  
Ergodic Theory ◽  
Elementary Proof ◽  
Nonstandard Analysis ◽  
Measure Theory ◽  
Difference Sets ◽  
Explicit Bound ◽  
Finite Set

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If $A$ and $B$ are sets of integers having positive upper Banach densities $a$ and $b$ respectively, then there exists a finite set $F$ of cardinality at most $1/ab$ such that $(A-B)+F$ covers arbitrarily long intervals.

scholarly journals A Problem of Ramanujan, Erdős, and Kátai on the Iterated Divisor Function

2011 ◽  
Vol 2012 (17) ◽  
pp. 4051-4061 ◽  
Author(s):  
Yvonne Buttkewitz ◽  
Christian Elsholtz ◽  
Kevin Ford ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Divisor Function
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