Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  609.10034
Autor:  Erdös, Paul; Pomerance, C.; Sárközy, A.
Title:  On locally repeated values of certain arithmetic functions. II. (In English)
Source:  Acta Math. Hung. 49, 251-259 (1987).
Review:  [For part I, cf. J. Number Theory 21, 319-332 (1985; Zbl 574.10012).]
It seems reasonable, in view of the Erdös-Kac theorem, to conjecture that the number of n \leq x with \nu(n) = \nu(n+1) is of exact order x(log log x)^-. The principal result of the paper is that there is a constant c such that |\nu(n)-\nu(n+1)| \leq c has at least order x(log log x)^- solutions n \leq x. It is stated that c = 3 is admissible. The proof uses the Erdös-Kac theorem to produce many pairs a,b with \nu(a) = \nu(b). One then takes a0,b0 to solve ab0-ba0 = 1, and uses the small sieve to find integers m for which \nu(am+a0), \nu(bm+b0) are both small. One may then take n = b(am+a0). The reviewer [Mathematika 31, 141-149 (1984; Zbl 529.10040)] showed by a different approach that the related equation d(n) = d(n+1) (where d(n) is the divisor function) has infinitely many solutions. It seems likely that one could handle \nu(n) = \nu(n+1) the same way, but as yet there seems no hope of "hybridizing" the two methods so as to solve the original conjecture.
In the final section of the paper, the number of solutions n \leq x of the equation \phi(n) = \phi(n+1), (where \phi(n) is Euler's function) is considered, and shown to be at most x/\exp((log x)1/3). The proof is closely related to C. Pomerance's treatment [J. Reine Angew. Math. 325, 183-188 (1981; Zbl 448.10007)] of amicable numbers.
Reviewer:  D.R.Heath-Brown
Classif.:  * 11N05 Distribution of primes
                   11N35 Sieves
                   11A25 Arithmetic functions, etc.
                   11A25 Arithmetic functions, etc.
Keywords:  sieve method; Erdös-Kac theorem; Euler's function
Citations:  Zbl 542.10037; Zbl 574.10012; Zbl 529.10040; Zbl 448.10007

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