Some Congruences for the Elementary Divisor Functions

1969 ◽  
Vol 76 (4) ◽  
pp. 395 ◽  
Author(s):  
D. B. Lahiri
Keyword(s):  
Elementary Divisor ◽  
Divisor Functions

Some Criteria for Hermite Rings and Elementary Divisor Rings

1974 ◽  
Vol 26 (6) ◽  
pp. 1380-1383 ◽  
Author(s):  
Thomas S. Shores ◽  
Roger Wiegand
Keyword(s):  
Triangular Matrix ◽  
Elementary Divisor ◽  
Diagonal Matrix ◽  
Finitely Generated ◽  
Elementary Divisor Ring ◽  
Hermite Ring

Recall that a ring R (all rings considered are commutative with unit) is an elementary divisor ring (respectively, a Hermite ring) provided every matrix over R is equivalent to a diagonal matrix (respectively, a triangular matrix). Thus, every elementary divisor ring is Hermite, and it is easily seen that a Hermite ring is Bezout, that is, finitely generated ideals are principal. Examples have been given [4] to show that neither implication is reversible.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Noncommutative elementary divisor rings

1988 ◽  
Vol 39 (4) ◽  
pp. 349-353
Author(s):  
B. V. Zabavskii
Keyword(s):  
Elementary Divisor

scholarly journals A PRODUCT FORMULA FOR COMBINATORIC CONVOLUTION SUMS OF ODD DIVISOR FUNCTIONS

2016 ◽  
Vol 38 (2) ◽  
pp. 243-257
Author(s):  
Kwangchul Lee ◽  
Daeyeoul Kim ◽  
Gyeong-Sig Seo
Keyword(s):  
Product Formula ◽  
Convolution Sums ◽  
Divisor Functions
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