The number of squarefree divisors of an integer

The infinitary divisors of a natural number n are the products of its divisors of the form pyÃŽÂ±2ÃŽÂ±, where py is a prime-power component of n and Ã¢ÂˆÂ‘ÃŽÂ±yÃŽÂ±2ÃŽÂ± (where yÃŽÂ±=0 or 1) is the binary representation of y. In this paper, we investigate the infinitary analogues of such familiar number theoretic functions as the divisor sum function, Euler's phi function and the MÃƒÂ¶bius function

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“…This is the classical divisor problem, discussed in [3].) For unitary divisors, we have the following result (obtained by Gioia and Vaidya [5]):…”

mentioning

confidence: 70%

Arithmetic functions associated with infinitary divisors of an integer

Cohen

Hagis

1993

International Journal of Mathematics and Mathematical Sciences

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“…It was slightly sharpened to O(x 1/2 ) in [3], but the exponent 1/2 remains unimproved to date. However, it can be further sharpened assuming the Riemann Hypothesis.…”

Section: Theorem 12 Let E(x) Denote One Of the Error Termsmentioning

confidence: 96%

Oscillations of a given size of some arithmetic error terms

Kaczorowski

Wiertelak

2009

Trans. Amer. Math. Soc.

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“…These functions are multiplicative and τ (r) (p a ) = 1, a r, 2, a > r. (1) σ (r) (p a ) = p a , a r, p a + p r , a > r. (2) In a special case of r = 0 we get well-studied unitary divisors. For example, it was proved in [3] that…”

Section: Introductionmentioning

confidence: 99%

“…(1) σ (r) (p a ) = p a , a r, p a + p r , a > r. (2) In a special case of r = 0 we get well-studied unitary divisors. For example, it was proved in [3] that (3)…”

Section: Introductionmentioning

confidence: 99%