CONTINUATION OF THE LAUDATION TO Professor Bui Minh Phong on his 65-th birthday

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1 Aales Uiv. Sci. Budapest., Sect. Comp. 47 (2018) CONTINUATION OF THE LAUDATION TO Professor Bui Mih Phog o his 65-th birthday by Imre Kátai (Budapest, Hugary) We cogratulated him i this joural (volume 38) five years ago. Durig the last five years may importat thigs has happeed with him. (1) The umber of gradchildre grew up to five. Their ames: David (2008), Mia (2014), Daiel (2016), Lili (2016) ad Victoria (2018). (2) He defeded his habilitatio thesis i 2013 (see [73]). (3) He has bee appoited as a full professor to the Computer Algebra Departmet of the Eötvös Lorád Uiversity. (4) Eötvös Lorád Uiversity hooured his promiet results i umber theory. (5) Bui Mih Phog has bee the vice presidet of Uio of Vietamese livig i Hugary durig the last twety years. Appreciatig his activity he had bee ivited to the 9th cogress of the Solidarity Uio (of Vietam) which had bee held i Oly 15 persos leavig outside of Vietam had bee ivited ad participated o this evet. He cotiued his research i umber theory. He wrote 34 ew papers i umber theory, some of them with coauthors. We elarge the categories to classify his ew results as follows: I. A characterizatio of the idetity with fuctioal equatios ([76], [82], [90], [92], [96], [102]). II. A characterizatio of additive ad multiplicative fuctios ([77], [78], [79],[80], [81],[88],[89],[91], [93], [94], [95], [97], [98], [99], [100], [101], [103]).

2 88 I. Kátai III. O additive fuctios with values i Abelia groups ([74], [104]). IV. A cosequece of the terary Goldbach theorem ([83]). V. O the multiplicative group geerated by { [ 2] } ([84], [85], [86], [87], [105]). I. A characterizatio of the idetity with fuctioal equatios A arithmetic fuctio g() 0 is said to be multiplicative if (, m) = 1 implies that g(m) = g()g(m) ad it is completely multiplicative if this relatio holds for all positive itegers ad m. Let M ad M deote the classes of complex-valued multiplicative, completely multiplicative fuctios, respectively. Let P, N be the set of primes, positive itegers, respectively. Let A, B N. We say that A, B is a pair of additive uiqueess sets (AU-sets) for M if oly oe f M exists for which f(a + b) = f(a) + f(b) for all a A ad b B, amely f() = for all N. I 1992, Spiro (J. Number Theory 42 (1992), ) showed that A = = B = P are AU-sets for M. For some geeralizatios of this result, we refer the works of J.-M. De Koick, I. Kátai ad B. M. Phog [38], B. M. Phog [60], B. M. Phog [66], K.-H. Idlekofer ad B. M. Phog [62], K. K. Che ad Y. G. Che (Publ. Math. Debrece, 76 (2010), ), J-H. Fag (Combiatorica, 31 (2011), ), A. Dubickas ad P. Šarka (Aequatioes Math. 86 (2013), o. 1-2, 81 89). I 1997, J.-M. De Koick, I. Kátai ad B. M. Phog [38] proved the followig theorem. Theorem A. ([38]) If f M with f(1)=1 satisfies f(p + m 2 ) = f(p) + f(m 2 ) for all p P, m N, the f() = for all N. For two completely multiplicative fuctios the followig theorem had bee proved. Theorem B. ([66]) If f, g M satisfy the equatios f(p + 1) = g(p) + 1 ad f(p + q 2 ) = g(p) + g(q 2 ) for all p, q P,

3 Cotiuatio of the laudatio to Professor Bui Mih Phog 89 the either f(p + 1) = f(p + q 2 ) = 0 ad g(π) = 1 for all p, q, π P or f() = g() = for all N. Fialy, B. M. Phog proved i 2016 the followig result: Theorem C. ([92]) Assume that f, g M with f(1)=1 satisfy f(p + m 2 ) = g(p) + g(m 2 ) ad g(p 2 ) = g(p) 2 for all primes p ad m N. The either for all primes p ad m N or for all p P,, m N. f(p + m 2 ) = 0, g(p) = 1 ad g(m 2 ) = 1 f() = ad g(p) = p, g(m 2 ) = m 2 I the other paper i 2016, B. M. Phog gave a complete solutio of multiplicative fuctios which satisfy some equatio. He proved: Theorem D. ([96]) Assume that o-egative itegers a, b with a + b > 0 ad f, g M satisfy the coditio f ( 2 + m 2 + a + b ) = g( 2 + a) + g(m 2 + b) for all, m N. Let S = g( 2 + a) ad A = (S 6 + 4S 5 S 3 S 2 3S 1 ). The the followig assertios are true: I. A {0, 1}. II. If A = 1, the g(m 2 + a) = m 2 + a, g(m 2 + b) = m 2 + b for all m N ad f() = for all N, (, 2(a + b)) = 1.

4 90 I. Kátai III. If A = 0, the there is a K {1, 2, 3} such that S +K = S for all N. All solutios of f ad g are give. Theorem E. ([102]) If f M satisfies the coditios f(p + m 3 ) = f(p) + f(m 3 ) for all p P, m N ad the f(π 2 ) = f(π) 2 for all π P, f() = for all N. II. A characterizatio of additive ad multiplicative fuctios We shall list some results amog papers [77], [79], [80], [88], [89], [94], [95], [97], [98], [99], [100], [101], [103]. I 1946, amog other results o additive fuctios, P. Erdős (A.Math., 47 (1946), 1 20) proved that if f A satisfies f( + 1) f() = o(1) as, the f() is a costat multiple of log. He stated several cojectures cocerig possible improvemets ad geeralizatios of his results. I additio P. Erdős cojectured that the last coditio could be weakeed to f( + 1) f() = o(x) as x. x This was later proved by I. Kátai i 1970 ad E. Wirsig i I 1981 I. Kátai (Studia Sci. Math. Hugar., 16 (1981), ) stated as a cojecture that if f A satisfies the coditio f( + 1) f() = o(1) as, the there is a suitable real umber τ such that f() τ log = 0 for all N. Improvig o earlier results of I. Kátai, E. Wirsig, Tag Yuasheg, Shao Pitsug i 1996 ad E. Wirsig, D. Zagier i 2001 showed that this cojecture is true. I 1981 I formulated the cojecture

5 Cotiuatio of the laudatio to Professor Bui Mih Phog 91 Cojecture 1. If g M ad the either 1 x 1 x g( + 1) g() = 0, x x g() 0, or g() = s, s C, Rs < 1. A cosequece of it is the Cojecture 2. If f A ad 1 x f( + 1) f() = 0, x the there is a suitable real umber τ for which f() τ log = h(), h() Z for all N. A. Hildebrad i 1988 proved that there exists a positive costat δ with the followig property. If f A satisfies f 0 ad f(p) δ for all primes p P, the if 1 x g( + 1) g() > 0. x It follows from this result that the above cojecture is true for additive fuctios satisfyig the coditio f(p) δ for all p P. The followig result improves the above theorem of Hildebrad. Theorem F. ([77]) Let k N be fixed. The there exists a suitable η(> 0) for which, if f j A (j = 0,..., k), ad f j (p) η for every p P, (j = 0,..., k), the if 1 x f 0 () f k ( + k) + Γ = 0 x implies that for every p P, p > k + 1, α N, we have furthermore f 0 (p α )... f k (p α ) 0 (mod 1), f j (p β ) f j (p β+1 ) (mod 1) if p β > k + 1, j = 0,..., k. Cosequetly, if f j A (j = 0,..., k), the Γ 0 (mod 1) ad f 0 ()... f k () 0 (mod 1) ( N).

6 92 I. Kátai Recetly, O. Klurma proved Cojecture 1. From his proof oe ca prove easily that if g M ad 1 g( + 1) g() = 0, log x x the either g() = s (s C, 0 Rs < 1) or 1 g() = 0. log x x I a joit paper of Idlekofer, Kátai ad Bui [97] we proved: If f M, x f( + 1) f() = O(log x), f() the either f() = s (s C, 0 Rs < 1) or x = O(log x). They proved furthermore: Let f M, f() = 0 whe (, K) > 1, (K is a fixed atural umber), ad sup sup 1 f() = log x x 1 f( + K) f() <, log x x the f() = s χ() (s C, 0 Rs < 1), where χ is a Dirichlet character (mod K). Recetly, Bui ad I proved the followig two theorems: Theorem G. ([100]) Let P (x) = a 0 + a 1 x + + a k x k, k {1, 2, 3}, f M, P (E)f() = a 0 f() + a 1 f( + 1) + + a k f( + k). Assume that x P (E)f() = O(log x) The either or x f() = O(log x) f() = s F () ( N), 0 Rs < 1

7 Cotiuatio of the laudatio to Professor Bui Mih Phog 93 ad P (E)F () = 0 for all. Theorem H. ([106]) Let f, g M, f()g() 0 ( N), A 0, ad that The ad 1 g(2 + 1) Af() = 0, log x x 1 f() > 0. log x x A = f(2), f() = δ+it ( N), 0 δ < 1, t R f() = g() for all odd. For some iteger q 2 let A q be the set of q-additive fuctios. Every N 0 ca be uiquely represeted i the form = a r ()q r with a r () {0, 1,, q 1}(= A q ) r=0 ad a r () = 0 if q r >. We say that f A q, if f : N 0 C f(0) = 0 ad f() = r=0 ( f a r ()q r) for all N. Let G = Z[i] be the set of Gaussia itegers. Kátai ad Szabó proved (Acta Sci. Math Szeged, 37 (1975), ) that if θ = A ± i (A N), the every γ G has a uique represetatio of the form γ = r 0 + r 1 θ + + r k θ k, where r j A θ = {0, 1,..., N 1}, N = A For some Gaussia iteger θ = A ± i (A N) let A θ be the set of θ-additive fuctios, that is f A θ, if f : G C, f(0) = 0 ad f(γ) = j=0 ( f r j θ j). A fuctio f : G C is completely multiplicative (f M (G)), if f(αβ) = f(α)f(β) for all α, β G.

8 94 I. Kátai Theorem I. ([95]) Let G = Z[i] be the set of Gaussia itegers. Assume that θ = A + i, A N ad f M (G) A θ. (A) γ G. If f(n) = f(θ)f(θ) 0, the either f(γ) = γ or f(γ) = γ for every (B) If f(n) = f(θ)f(θ) = 0, the f(γ) = f(c), where c A θ, γ c (mod N). Theorem J. ([99]) Assume that f M, g A 2, h A 3 satisfy the coditio The f() = g() + h() for all N. f(1) = g(1) + h(1) = 1. For every N, oe of the followig assertios hold: f() = χ 2 (), g() = h(1) + χ 2 (), h() = h(1), where χ 2 is the Dirichlet-character (mod 2), or f() = χ 3 (), g() = g(1), h() = g(1) + χ 3 (), f() =, g() = g(1), h() = h(1). The solutios of all those pairs f M, g A θ, for which f() = g 2 () for all N are give as follows: Theorem K. ([101]) Assume that f M, g A θ satisfy the coditio The either f() = g 2 () for all N. f(q 0 ) = 0, q 0 q ad f() = g 2 () = χ q0 (), where χ q0 is a Dirichlet character (mod q 0 ), or f() = 2 for all N, ad either g() = ( N), or g() = ( N). III. O additive fuctios with values i Abelia groups I formulated the ext cojectures. Cojecture 3. If f 0, f 1,..., f k A ad f 0 () + f 1 ( + 1) + + f k ( + k) 0, as,

9 Cotiuatio of the laudatio to Professor Bui Mih Phog 95 the there are τ 0,..., τ k R such that τ τ k = 0 ad f 0 () τ 0 log = = f k () τ k log = 0 for all N. Cojecture 4. If f 0, f 1,..., f k A ad f 0 () + f 1 ( + 1) + + f k ( + k) Z for all N, the f j () Z for all N ad j = 0, 1,..., k. Cojecture 2 is kow for k = 2, 3 It is proved i 2012 by K. Chakraborty, I. Kátai ad B. M. Phog that Cojecture 2 is true for the case k = 4 by assumig that the relatio holds for all Z. f 0 () + f 1 ( + 1) + f 2 ( + 1) + f 3 ( + 1) + f 4 ( + 1) Z Theorem L. ([74]) If 0 are Abelia groups, Γ, {f 0, f 1, f 2, f 3, f 4, f 5 } A ad f 0 () + f 1 ( + 1) + f 2 ( + 2) + f 3 ( + 3) + f 4 ( + 4) + f 5 ( + 5) + Γ 0 is true for all Z, the Γ 0 ad f j () 0 for all Z, j {0, 1,, 5}. IV. A cosequece of the terary Goldbach theorem H. A. Helfgott proved i 2013 that every odd iteger larger tha 5 ca be writte as the sum of three primes. We deduced from this the followig theorem. Theorem M. ([83]) Let k 3, k fix, f : N C, g : P C. Assume that f (p 1 + p p k ) = g(p 1 ) + g(p 2 ) + + g(p k )

10 96 I. Kátai holds for every p 1, p 2,, p k P. The there exist suitable costats A, B C such that (2.2) f() = A + kb for all N, 2k ad g(p) = Ap + B for all p P.) V. O the multiplicative group geerated by { [ 2] } Let α, β be distict positive real umbers ad let Q + be the multiplicative group of the positive ratioals. I [84], we started the followig cojecture. Cojecture 1. Let ξ = [α] [β] ( N), F α,β be the multiplicative group geerated by {ξ N}. If α, β are distict real umbers at least oe of which is irratioal, the F α,β = Q +. We proved this cojecture i the special case α = 2, β = 1 ad proved some theorems: ([84]) Assume that f A ad ( f [ ) 2] f() C ( ). The f() = A log ( N), where A = 2C log 2. ([85]) Let f, g A, C R, δ() = g([ 2]) f() C. Assume that 1 { } x x δ() > ɛ = 0 holds for every ɛ > 0. The f() = g() = A log, where A = 2C ([86]) Let ε() 0. Assume that ε() log log 2 <. =2 Let f, g M, C C, f() = g() = 1 ( N) ad g([ 2]) Cf() ε() ( N). log 2. The f() = g() = iτ (τ R), where C = ( 2) iτ. ([86]) If f, g M, C C, C 0, f() = g() = 1 ( N) ad g([ 2]) Cf() <, =1 the f() = g() ( N), furthermore C 2 = f(2).