* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download On distribution of arithmetical functions on the set prime plus one
List of prime numbers wikipedia , lookup
History of trigonometry wikipedia , lookup
Big O notation wikipedia , lookup
Functional decomposition wikipedia , lookup
Vincent's theorem wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Mathematical proof wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Law of large numbers wikipedia , lookup
History of the function concept wikipedia , lookup
Dirac delta function wikipedia , lookup
Continuous function wikipedia , lookup
Karhunen–Loève theorem wikipedia , lookup
Fermat's Last Theorem wikipedia , lookup
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Nyquist–Shannon sampling theorem wikipedia , lookup
Series (mathematics) wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Non-standard calculus wikipedia , lookup
C OMPOSITIO M ATHEMATICA I. K ÁTAI On distribution of arithmetical functions on the set prime plus one Compositio Mathematica, tome 19, no 4 (1968), p. 278-289 <http://www.numdam.org/item?id=CM_1968__19_4_278_0> © Foundation Compositio Mathematica, 1968, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On distribution of arithmetical functions on the set prime plus one by I. Kátai 1. Introduction P. Erdôs proved the following theorem [1]. Let f(n) be a real valued additive number-theoretical function, and put Put Then the distribution-functions FN(x) tend for N - + co to a limiting distribution function F(x) at all points of continuity of F(x), if the following three conditions are satisfied: It has been shown also by P. Erdôs that F(x) is continuous if and only if the series 03A3f(p) ~ 0 1/p diverges. New proof of this theorem has been given by H. Delange [2] and by A. Rényi [3]. A multiplicative function g(n) is called strongly multiplicative, if for all primes p and all positive integers k it satisfies the condition H. Delange proved the following theorem [4]. 278 279 If g(n) is a strongly multiplicative number-theoretical function such that Ig(n)1| 1 for n 1, 2, ..., and such that the series = converges, then exists and new proof of this theorem has been given by A. Rényi [5]. Throughout the paper p, q denote primes, and 03A3p and Wp denote a sum and a product, respectively, taken over all primes. Let further li x = ~x2 du/log u. The aim of this paper is to prove the following statement. A THEOREM 1. Let g(n) be a compler-valued multiplicative function 1 f or n and such that the series such that|g(n)| 1, 2, = converges. Let N(g) denote the ..., product Then From this theorem easily follows the THEOREM 2. Let f(n) be a real valued additive number-theoretical which satisfies the conditions 1, 2, 3, of the theorem of Erdös. Put function distribution-functions Fv(y) tend for N ~ oo to a lintiting distribvtron-function F(y) at all points of continuity of F(y). Further F(y) is a continuous function if and only if Then the 280 2. Deduction of Theorem 2 from Theorem 1 In what follows c, cl , C2’ denote constants not always the in different places. For the proof of Theorem 2 we need to prove only that the sequence of characteristic functions ... same converges to a function ~(u), which is a continuous one on the real axis. It is easy to verify that from the conditions 1/2/3 it follows that converges for every real u. Using now Theorem 1 with obtain that q;N(U) - ~(u), where g(n) = eiuf(n) we The continuity of (2.3) is guaranteed by the continuity of which follows from the conditions 1/2/3 evidently. For the proof of the continuity of F(x) in the case (2.2), remark the following. P. Levy proved the following theorem [8]. Let X1, X2, ..., X n , be a sequence of independent random variables with discrete distribution and suppose that there exists the sum we ... with probability 1. Let Then the distribution function of X is continuous if and only if 281 Let now the Xf)’s be teristic functions independent random variables with charac- and let It is evident from (2.3) that X has the distribution function F(x), and so this is continuous if and only if 3. The proof of Theorem 1 We need the following Lemmas. LEMMA 1. Let g(p) be a complex-valued function defined primes, for zvhich ig(p)j 1 and on the converges. Then for every positive constant c, further and PROOF. The assertion in (3.3) since (3.4) is an immediate consequence of further 03A3x1/2px 1/p is bounded, and the second sum on the hand side tends to zero because of (3.3). Let us put right 282 From the convergence of (3.1) it follows that converges too. This |arg g(p)| >c follows. From the The first sum has positive terms and the inequality involves that 1-Re g(p) > c1(>0). Hence (3.2 ) inequality |a+bi|2 ~ 2(laI2+ Ib12) sum on the right hand side it follows that evidently converges since It is sufficient to prove that Using the inequality we have (3.5). LEMMA 2. Let Nk(x) denote the number in primes p, of solutions of the equation q. Then f or k x, where c is an absolute constant. For the proof see Prachar’s book [6], Theorem 4.6, p. 51. Let n(x, k, l) denote the number of primes in the arithmetical progression ~ l (mod k) not exceeding x. LEMMA 3. (Brun-Titchmarsh). where ca is a constant depending For the proof see [6J. For all k on c5 only. x1-03B4, 03B4 > 0 283 LEMMA 4. (E. Bombieri [7]). xi (log x)-B; B > 2A + 23, A arbitrary constant. where Y the number of divisors of n. be Let i(n) = LEMMA 5. where c is a constant. The proof is very simple and so can be omitted. Let us define the multiplicative function gK(n) by By other words Let us we put for any natural number n put further divisors of n and 03BC(n) is the hK(n) is also a multiplicative function, gK(p03B1)-gK(p03B1-1); hK(p) 0 for p > K; hK(p) = 0 for where d runs over all Môbius function. Then hK(p03B1) p03B1-1 > K, = us (positive) = oc ~ 2. Let further Let putting h(n) be defined by introduce the notation Choose now K1 (1 4 - 03B5) log x, .K2 x1 4, and 03B4 are suitable small positive numbers. We shall prove the following relations: = = Ka = ael-8, where B 284 Theorem 1 follows if we choose 03B4 = 03B4(x) tending to slowly that the right hand side of (3.8) is o(li x). First we prove (3.6). We have zero so where Using the prime number theorem, for d > ae!-8/2 because we obtain that hK1(d) if x is sufficiently large. Since ig(n)l 1, so |hK1(p03B1)| ~ 2 and IhKl(d)1 S r(d). For the estimation of R, we split all of the d’s, d into two classes H1, 212 as follows: d belongs to 3ti or H2 according to that 03C4(d) ~ (log a(d) > (log x)5, respectively. Using Lemma 3 and Lemma 5 we sum not exceed Hence 0 ael-8/2 x)5 or have Otherwise, using the Bombieri’s result (Lemma 4), that the = we have 285 Further we have From the convergence of the series 03A3 (g(p) - 1)/p it follows that the product on the right hand side tends to N(g) for x ~ + ~. So (3.6) is proved. Let now g(n) be a multiplicative function defined by It is evident that g(n) g(n) except eventually those n’s for which there exists a q, q > Kl , such that q2Bn. So = From (3.10) follows. Using the formulas if all primdivisor q of p+1 in the interval K, q K2 satisfies the relation larg g(q)1 set of the p’s the Let denote H3 n/2. this the and other p’s. possessing property, H4 We can easily estimate the sum since 286 and by (3.2 ) Let From (3.11) Using (3.3) we have that in Lemma 1 and Lemma 3 Further, from the Cauchy’s inequality Using Bombieri’s result since So we whence proved that (3.7) follows. Similarly we have we have that we have 287 Using Lemma 3 and Further using (3.3) (3.4) and in Lemma 1 we have that obtain that (3.2) we 2 have Hence (3.8) follows. Finally using Lemma we because So the inequality theorem follows. (3.9) is proved, and from (3.6 )-(3.9) our 4. Some remarks 1. From positive our valued that then putting Theorem 2 it follows evidently that if g(n) is a multiplicative number-theoretical function such 288 the distribution functions FN(y) tend for N - +~ to a distribution function F(y) at all points of continuity of Hence it follows especially that the functions (a(n) denotes the sum of the divisors of tribution functions. n) have limiting F(y). limiting dis- 2. Recently M. B. Barban, A. I. Vinogradov, B. V. Levin proved that all results of J. P. Kubilius theory (see [9]) are valid for strongly additive arithmetic functions belonging to the class H, when the argument runs through "shiffed" primes {p-l}, (see [10], [ll] ). REFERENCES P. [1] ERDÖS, On the 13 H. [2] A. [3] H. [4] density of some (1938), 1192014127. sequences of numbers, III. J. London Math. Soc. DELANGE, Un theorème sur les fonctions arithmétiques multiplicatives et ses applications, Ann. Sci. Ecole Norm. Sup. 78 (1961), 1-29. RÉNYI, On the distribution of values of additive number-theoretical Math. Debrecen 10 (1963), 2642014273. functions, Publ. DELANGE, Sur les fonctions arithmétiques (1961), 2732014304. multiplicatives, Ann. Sci. Ecole Norm. Sup. 78 A. [5] K. RÉNYI, A new proof of 3232014329. a theorem of Delange, Publ. Math. Debrecen, 12 PRACHAR, Primzahlverteilung, Springer Verlag 1957. E. BOMBIERI, [7] On the large sieve, Mathematika, 12 (1965), 201-225. P. LEVY, [8] Studia Mathematica, 3 6 (1931), p. 150. J. P. KUBILIUS, [9] Probabilistic methods in number theory, Vilnius, 1962 (in Russian). [6] (1965), 289 M. B. BARBAN, [10] Arithmetical functions on "thin" sets, Trudi inst. math. im. V. I. Romanovskovo, Teor. ver. i math. stat. vüp 22, Taskent, 1961 (in Russian). M. B. BARBAN, A. I. VINOGRADOV, and B. V. LEVIN, [11]Limit set of laws for arithmetic functions of J. P. Kubilius class H, defined on the "shiffed" primes, Litovsky Math. Sb. 5 (1965), 120148 (in Russian). (Oblatum 22-3-’67).