Special arithmetic functions connected with the divisors, or with the digits of a number
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SPECIAL ARITHMETIC FUNCTIONS CONNECTED WITH THE DIVISORS, OR WITH THE DIGITS OF A NUMBER
4.1
Introduction
There are many particular arithmetic functions, numbers or sequences connected with some important notions or results which appear in Number theory, and in fact all Mathematics. Some of them are non standard functions (such as d, σ, ϕ, µ, ω, , p, P etc.), and were studied in [260] or in former chapters of this book. The aim of this chapter is the study of some other functions which at one part are not so well-known, and are scattered in various fields of study, or at another part have not been previously included. The former one includes arithmetic functions connected to the prime factorization of a number, or the consecutive divisors (prime or not). The functions related to the digits of a number written e.g. in a decimal (or binary) scale constitute also an important field of study, with many applications. The divisors and the digits of a number have a strong connection, it is sufficient to only mention the congruence property n ≡ s(n) (mod 9), where s(n) denotes the sumof-digits of n in decimal representation of n. 329
CHAPTER 4
4.2 1
Special arithmetic functions connected with the divisors of a number Maximum and minimum exponents
Let n = p1a1 . . . prar > 1 be the prime factorization of n, and put H (n) = max{a1 , . . . , ar },
h(n) = min{a1 , . . . , ar }
(1)
and H (1) = h(1) = 1. These functions (called as the maximum and minimum exponents in factoring) occured also in Chapter III, when extending the Euler divisibility theorem. In 1969 I. Niven [265] proved that H (n) = c0 x + R H (x) (2) n≤x ∞ where c0 = 1 + (1 − ζ (k)−1 ), and R H (x) = o(x) and that k=2
h(n) = x + Rh (x)
(3)
n≤x
√ √ where Rh (x) = c x + o( x), with c = ζ (3/2)/ζ (3) (where ζ is the Riemann zeta function) conjectured previously by P. Erd¨os (see [265]). We note here that the generating functions of max{n 1 , . . . , n k } and min{n 1 , . . . , n k } (where a1 , . . . , ak are arbitrarily nonnegative integers) are deduced by L. Carlitz [52]. For example ∞ n 1 ,...,n k =0
min{n 1 , . . . , n k }x1n 1 . . . xkn k =
x1 x2 . . . xk (1 − x1 ) . . . (1 − xk )(1 − x1 . . . xk )
(4)
D. Suryanarayana and R. Sitaramachandra Rao [322] improved Niven’s results to
and
R H (x) = O(x 1/2 exp(−c(log x)3/5 (log log x)−1/5 ))
(5)
√ √ √ √ Rh (x) = c x + c1 3 x + c2 4 x + c3 5 x +O(x 1/6 )
(6)
A(x)
where c > 0 and c1 , c2 , c3 are constants. H. Cao [49] rediscovered (5), and a weaker form of (6). In another paper [50] he improved the exponent 3/5 to 3/5 − ε. In [163] T. Gu and H. Cao assert without proof that Rh (x) = A(x) + C4 x 1/6 + O(x 1/6 exp(−D(log x)4/7 (log log x)−3/7 ) 330
(7)
SPECIAL ARITHMETIC FUNCTIONS
but, according to A. Ivi´c [191] this was formerly established. These results can be further improved by using the strongest known form for the asymptotic formula on the number of squarefull integers not exceeding x. ˇ Porubsk´y [278] has extended (2) and a By applying the ideas used by Niven, S. wea
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