Book
The Distribution of Totients: The distribution of totients
| Τίτλος: | The Distribution of Totients: The distribution of totients |
|---|---|
| Συγγραφείς: | Kevin Ford |
| Πηγή: | Developments in Mathematics ISBN: 9781441950581 |
| Publication Status: | Preprint |
| Στοιχεία εκδότη: | Springer US, 1998. |
| Έτος έκδοσης: | 1998 |
| Θεματικοί όροι: | normal number of prime factors, sum of divisors function, prime \(k\)-tuples conjecture, Mathematics - Number Theory, multiplicities, Arithmetic functions, related numbers, inversion formulas, Sierpiński's conjecture, 0102 computer and information sciences, 01 natural sciences, Other results on the distribution of values or the characterization of arithmetic functions, Sierpinski's conjecture, Euler's function, totients, FOS: Mathematics, Asymptotic results on arithmetic functions, Number Theory (math.NT), 0101 mathematics, multiplicative arithmetic functions, Carmichael's conjecture |
| Περιγραφή: | This paper is an announcement of many new results concerning the set of totients, i.e. the set of values taken by Euler’s ϕ \phi -function. The main functions studied are V ( x ) V(x) , the number of totients not exceeding x x , A ( m ) A(m) , the number of solutions of ϕ ( x ) = m \phi (x)=m (the “multiplicity” of m m ), and V k ( x ) V_{k}(x) , the number of m ≤ x m\le x with A ( m ) = k A(m)=k . The first of the main results of the paper is a determination of the true order of V ( x ) V(x) . It is also shown that for each k ≥ 1 k\ge 1 , if there is a totient with multiplicity k k , then V k ( x ) ≫ V ( x ) V_{k}(x) \gg V(x) . We further show that every multiplicity k ≥ 2 k\ge 2 is possible, settling an old conjecture of Sierpiński. An older conjecture of Carmichael states that no totient has multiplicity 1. This remains an open problem, but some progress can be reported. In particular, the results stated above imply that if there is one counterexample, then a positive proportion of all totients are counterexamples. Determining the order of V ( x ) V(x) and V k ( x ) V_{k}(x) also provides a description of the “normal” multiplicative structure of totients. This takes the form of bounds on the sizes of the prime factors of a pre-image of a typical totient. One corollary is that the normal number of prime factors of a totient ≤ x \le x is c log log x c\log \log x , where c ≈ 2.186 c\approx 2.186 . Lastly, similar results are proved for the set of values taken by a general multiplicative arithmetic function, such as the sum of divisors function, whose behavior is similar to that of Euler’s function. |
| Τύπος εγγράφου: | Part of book or chapter of book Article |
| Περιγραφή αρχείου: | application/xml |
| ISSN: | 1572-9303 1382-4090 |
| DOI: | 10.1007/978-1-4757-4507-8_8 |
| DOI: | 10.1023/a:1009761909132 |
| DOI: | 10.1090/s1079-6762-98-00043-2 |
| DOI: | 10.48550/arxiv.1104.3264 |
| Σύνδεσμος πρόσβασης: | http://arxiv.org/abs/1104.3264 https://zbmath.org/1149588 https://doi.org/10.1090/s1079-6762-98-00043-2 https://link.springer.com/article/10.1023/A:1009761909132 https://experts.illinois.edu/en/publications/the-distribution-of-totients |
| Rights: | Springer Nature TDM arXiv Non-Exclusive Distribution URL: https://www.ams.org/publications/copyright-and-permissions |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....bf7ad7e313f4afcef3f6aaebc218c6df |
| Βάση Δεδομένων: | OpenAIRE |
| ISSN: | 15729303 13824090 |
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| DOI: | 10.1007/978-1-4757-4507-8_8 |