Arithmetical functions, sets of uniqueness, generalized number systems  Page description

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Details of project

  
Identifier
46993
Type K
Principal investigator Kátai, Imre
Title in Hungarian Aritmetikai függvények, egyértelműségi halmazok, általánosított számrendszerek
Title in English Arithmetical functions, sets of uniqueness, generalized number systems
Panel Mathematics and Computing Science
Department or equivalent Department of Computer Algebra (Eötvös Loránd University)
Participants Bui Minh, Phong
Farkas, Gábor
Fehér, János
Fülöp, Ágnes
Kovács, Attila
Tóth, László
Starting date 2004-01-01
Closing date 2007-12-31
Funding (in million HUF) 5.850
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
Az alábbi területen sikerült érdekes eredményeket elérni: 1. Additív függvények eloszlása, multiplikatív függvények középértéktétele rövid intervallumokon. Ramachandra Hooley-Huxley kontúrra vonatkozó eredmények alkalmazása. 2. $q$-additív függvények értékeloszlása különböző feltételeket kielégítő részhalmazokon. 3. Azoknak a $q$-multiplikatív függvényeknek a karakterizálása, amelyek a prímszámok halmazán egy adott függvényosztályhoz ($\Cal L^\alpha, \Cal L^*$ tartoznak). 4. Egyértelműségi és $\mod 1$ egyértelműségi halmazok. 5. A Daboussi-féle problémakör. 6. Az Euler-féle $\varphi$ függvény és iteráltjai. 7. Az Erdős-Wintner tétel analogonja az $\{n+1|n\leq x,\ \omega(n)=k\}$ halmazra, ahol $\omega(n)$ az $n$ prímosztóinak a száma. Közlésre elfogadott cikkek: 1. J.-M. De Koninck and I. Kátai, On the local distribution of $\omega(GCD(n,\varphi_k(n)))$, Canadian Math. Bull 2. J.-M. De Koninck and I. Kátai, On an estimate of Kanold, International Journal of Mathematics and Analysis 3. I. Kátai, On $q$-additive and $q$-multiplicative functions, Conference in Allahabad, 2006 December 4. K.-H. Indlekofer and I. Kátai, Some remarks on trigonometric sums, Acta Math. Hung. 5. J.-M. De Koninck, N. Doyon and I. Kátai, Counting the number of twin Niven numbers, Ramanujan Journal 6. I. Kátai and M.V. Subbarao, Distribution of additive and $q$-additive functions under some conditions II., Publ. Math. Debrecen
Results in English
Interesting and important results have been proved in the following topics: 1. Distribution of additive functions on short intervals. 2. Mean-values of multiplicative functions on short intervals. Application of the method of Ramachandra concerning the Hooley-Huxley contour. 3. Distribution of linear combination of $q$-additive functions on some arithmetically characterized subsets of integers. 4. Characterization of those $q$-multiplicative functions which are defined on the set of primes and belong to some classes of functions ($\Cal L^\alpha,\Cal L^*$). 5. Sets of uniqueness and sets of uniqueness $\mod 1$. 6. Some analogons of the theorem of Daboussi.
Full text http://real.mtak.hu/1655/
Decision
Yes





 

List of publications

  
K.-H. Indlekofer and I. Kátai: On the normal order of $\\varphi_{k+1}(n)/\\varphi_k(n)$, where $\\varphi_k$ is the $k$-fold iterate of Euler’s functions, Liet. Matem. rynk 44, 68-84, 2004
I. Kátai and M.V. Subbarao: On the local distribution of the iterated divisor function, Mathematica Pannonica 15, 2004
I. Kátai and M.V. Subbarao: Distribution of additive and q-additive functions under some conditions, Publ. Math. Debrecen 64, 167-187, 2004
K.-H. Indlekofer and I. Kátai: On the normal order of $\\varphi_{k+1}(n)/\\varphi_k(n)$, where $\\varphi_k$ is the $k$-fold iterate of Euler’s functions, Liet. Matem. rynk 44, 68-84, 2004
K.-H. Indlekofer and I. Kátai: On the normal order of $\\varphi_{k+1}(n)/\\varphi_k(n)$, where $\\varphi_k$ is the $k$-fold iterate of Euler’s functions, Liet. Matem. rynk 44, 68-84, 2004
I. Kátai and M.V. Subbarao: On the function $\xi(s)\xi(s-a)...\xi(s-ra)$, Math. Pannonica, vol. 16, 19-27, 2005
I. Kátai: Some results and problems on arithmetical functions, Proc. Intern. Conf. - Number Theory, No 1., (2004), 39-50; Lecture Notes Series, Ramanujan Math. Soc., 2005
J.M. De Koninck and I. Kátai: On the mean value of the index of composition of an integer, Monatshefte Math. 145, 131-144, 2005
J.M. De Koninck and I. Kátai: On the multiplicative group generated by shifted binary quadratic forms, Annales Univ. Budapest Sectio Math., 47 (2004), 17-28, 2005
J. Fehér and I. Kátai: On sets of uniqueness for additive and multiplicative functions over the multiplicative group generated by the polynomial $x^2+a$, Annales Univ. Budapest Sectio Math., 47 (2004), 143-156, 2005
I. Kátai and M.V. Subbarao: On a new proof of a theorem of Indlekofer and Timofeev, Annales Univ. Budapest Sectio Math., 47 (2004), 29-33, 2005
I. Kátai: On the number of prime divisors of the iterates of the Carmichael function, Math Pannonica, 16., 205-209, 2005
I. Kátai: Square-free values of the Carmichael function, Math Pannonica, 16/2., 199-203, 2005
I. Kátai and M.V. Subbarao: Distribution of $2$-additive functions under some conditions, Annales Univ. Sci. Budapest, Eötvös, Sect. Comp., 26, 137-143., 2006
I. Kátai and M.V. Subbarao: Some further remarks on the $\varphi$ and $\sigma$ functions, Annales Univ. Sci. Budapest, Eötvös, Sect. Comp., 26, 51-63., 2006
I. Kátai and M.V. Subbarao: A remark on the product partition of integers into $k$ parts, Annales Univ. Sci. Budapest, Eötvös, Sect. Comp., 26, 43-49., 2006
I. Kátai and M.V. Subbarao: A remark on product partition, Acta Math. Hung., 111(4), 305-310., 2006
I. Kátai: A remark on a paper of Luca, Acta Math. Hung., 113(4), 313-318., 2006
J.M. De Koninck and I. Kátai: On the local distribution of certain arithmetical functions, Liet. Matem. Rink. 46(3), 315-333., 2006
I. Kátai: A remark on trigonometric sums, Acta Math. Hung. 112, 221-225., 2006
I. Kátai and M.V. Subbarao: On the maximal and minimal exponent of the prime divisors of integers, Publ. Math. Debrecen, 68, 477-488., 2006
I. Kátai: Research problems in number theory III., Analytic Number Theory and Surrounding Areas, RIMS Kokyuroku 1511, Auguot 2006, Kyoto Japan, 121-128., 2006
I. Kátai and M.V. Subbarao: A remark on a paper written by J.M. De Koninck and A. Ivic, Mathematica Pannonica, 17/2., 2006
J.-M. De Koninck and I. Kátai: Distribution of arithmetical functions on certain subsets of integers, Rocky Mountain Journal of Mathematics 37/5, 1459-1482, 2007
I. Kátai and M.V. Subbarao: On product partitions and asymptotic formulas, The Riemann zeta function and related themes: Lecture Notes Series 2, Ramanujan Math. Soc. 99-114, 2006
I. Kátai and M.V. Subbarao: The distribution of integers with given number of prime factors in almost all short intervals, The Riemann zeta function and related themes: Lecture Notes Series 2, Ramanujan Math. Soc., 115-120, 2006
I. Kátai: On the linear combination of $q$-additive functions at prime places, Acta Math. Hung. 117, 361-372, 2007
I. Kátai: On the linear combination of $q$-additive functions at prime places, Acta Math. Hung. 117, 361-372, 2007
J.-M. De Koninck, I. Kátai and M.V. Subbarao: On the index of composition of integers from various sets, Archiv der Math. 88, 524-536, 2007
I. Kátai: Distribution of $q$-additive functions on the set of primes, Liet Matem. rynk 47, 29-38, 2007
J.-M. De Koninck and I. Kátai: Sums of reciprocals of additive functions running over short intervals, Colloquium Math. 107, 317-326, 2007
J.-M. De Koninck, I. Kátai and M.V. Subbarao: A consequence of a theorem of Filaseta, Annales des Sciences Mathematiques du Quebec, 30, 55-62, 2006
L. Germán and I. Kátai: Distribution of $q$-additive functions on the set of integers having $k$ prime factors, Annales Univ. Sci. Budapest, Eötvös Section Comp., 27, 65-74, 2007
J. Fehér and I. Kátai: Some remarks on sets of uniqueness for additive and multiplicative functions, Annales Univ. Sci. Budapest, Eötvös Section Comp., 27, 57-63, 2007
I. Kátai: On the average prime divisors, Annales Univ. Sci. Budapest, Eötvös Section Comp., 27, 137-144, 2007
I. Kátai: On a problem of A. Ivic, Mathematica Pannonica, 18/1, 11-18, 2007
L. Germán: The distribution of an additive arithmetical function on the set of shifted integers having $k$ distinct prime factors, Annales Univ. Sci. Budapest, Eötvös Section Comp., 27, 187-215, 2007
B. M. Phong: On integer-valued arithmetical functions satisfying congruence properties, Annales Univ. Sci. Budapest, Eötvös Section Comp., 27, 229-244, 2007
I. Kátai and M.V. Subbarao: Some remarks on the $\varphi$ and $\sigma$ functions, Annales Univ. Sci. Budapest, Eötvös Section Comp., 25, 113-130, 2005
J.-M. De Koninck and I. Kátai: On the average of $d(n)\omega(n)$ and similar functions on short intervals, Annales Univ. Sci. Budapest, Eötvös Section Comp., 25, 131-142, 2005
I. Kátai and M.V. Subbarao: Distribution of $q$-additive functions on some subsets of integers, Math. Pannonica, 18/1, 11-18, 2007



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