G. Sburlati RANDOM VARIABLES RELATED TO DISTRIBUTIONS OF SUMS MODULO AN INTEGER

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1 Rend. Sem. Mat. Univ. Pol. Torino Vol. 60, 3 (2002) G. Sburlati RANDOM VARIABLES RELATED TO DISTRIBUTIONS OF SUMS MODULO AN INTEGER Abstract. Starting from known results concerning distributions of sums modulo an integer, we build two random variables, ξ (E,v) (with = E P, v N, v 3, P denoting the set of prime numbers) and η (E,v) (with = E P\{2}, v N, v 2) and find upper bounds for their expected values; the probability functions related to such random variables are also shown.. The functions l E and L E (E P, E = ) There is a wide literature about congruence equations (see, for example, []) and in the last twenty years interesting related formulas and functions have been derived: among these, expressions giving the number of solutions of linear congruences. Counting such solutions has also nice relations with statistic and probabilistic problems like the distribution of the values taken by particular sums in Z r (r N), as we are going to see. Two arithmetic functions l E and L E, labelled by a generic non-empty subset E of the set P of prime numbers, are known (see [4]; for a detailed discussion of the particular case E = P\{2} see [3]) with the following properties. For given positive integers k and r, with all prime divisors of r lying in E, we set D = {d N : d r}, consider two generic elements = (d, d 2,..., d k ) D k and H = (h, h 2,..., h k ) (Z r ) k and define the function { S = S (H, ) r,k : D D 2... D k Z r (x, x 2,..., x k ) k j= h j x j (mod r), with D j = {x Z r : (x, r) = d j } for j =, 2,..., k. For any fixed positive integer v we denote by S v the set of functions S (each of which corresponds to a particular choice of r, k, H, ) such that for every prime p D we have ({ j, j k : p/ h j d j }) v. For any S S v and a Z r, the integer N S,a denotes the number of solutions of the congruence equation S(x, x 2,..., x k ) a (mod r) (formulas for N S,a in particular cases can be found in [2], [5] and [6]), while for each S we pose N S = r a Z r N S,a. The numbers l E = l E (v) and L E = L E (v) are then defined as the lower and the upper bounds, respectively, of the ratio N S,a N S for S ranging over S v 57

2 58 G. Sburlati and a over Z r. The following equalities have been proven in [4]: [ [ + l E (v) = (p ) s(v) ] ; L E (v) = (p ) t(v) where, for each v, s(v) and t(v) are, respectively, the greatest even integer and the greatest odd integer not higher than v. ], 2. The random variable ξ (E,v) ( = E P, v N, v 3) After fixing a non-empty subset E of P and an integer v 3, for each p E we can consider the number L {p} (v) = + and define the function (p ) t(v) { ξ(e,v) : E R p [ln L {p} (v)]/[]. The following property holds. PROPOSITION. The function ξ (E,v) is a random variable over the set E. Proof. For any p E we have L E (v) L {p} (v) > ; it follows that p E, ξ (E,v) (p) = [ln L {p} (v)]/[] > 0. Furthermore, the following equalities are verified: ln L {p} (v) ln L {p} (v) ξ (E,v) (p) = = =. Then ξ (E,v) is a random variable over E. A function Pr (ξ,e,v), related to ξ (E,v), can be defined with range equal to the set of all subsets of E, by posing I E, Pr (ξ,e,v) (I) = p I ξ (E,v) (p), where the empty sum is considered equal to 0. The following relations are satisfied:. I E, Pr (ξ,e,v) (I) 0; 2. I, J E with I J =, Pr (ξ,e,v) (I J) = Pr (ξ,e,v) (I) + Pr (ξ,e,v) (J); 3. Pr (ξ,e,v) (E) =.

3 Random variables and congruences 59 Then the function Pr (ξ,e,v) is a probability over the set E. EXAMPLE. Let us take E = {2, 3, 5, 7} and v = 3. We have: L {2} (3) = + (2 ) 3 = 2; L {3}(3) = + (3 ) 3 = 9 8 ; L {5} (3) = + (5 ) 3 = ; L {7}(3) = + (7 ) 3 = ; L E (3) = From such equalities we deduce that: ξ (E,3) (2) = ξ (E,3) (5) = L {p} (3) = = ln 2 ln(405/644) 0.834; ξ ln(9/8) (E,3)(3) = ln(405/644) 0.47; ln(65/64) ln(405/644) 0.087; ξ (E,3)(7) = ln(27/26) ln(405/644) The related probability function Pr (ξ,e,3) is defined as follows: Pr (ξ,e,3) ( ) = 0; Pr (ξ,e,3) ({2}) = ξ (E,3) (2) 0.834; Pr (ξ,e,3) ({3}) = ξ (E,3) (3) 0.47; Pr (ξ,e,3) ({5}) = ξ (E,3) (5) 0.087; Pr (ξ,e,3) ({7}) = ξ (E,3) (7) ; Pr (ξ,e,3) ({2, 3}) = ξ (E,3) (2) + ξ (E,3) (3) ; Pr (ξ,e,3) ({2, 5}) = ξ (E,3) (2) + ξ (E,3) (5) ; Pr (ξ,e,3) ({2, 7}) = ξ (E,3) (2) + ξ (E,3) (7) ; Pr (ξ,e,3) ({3, 5}) = ξ (E,3) (3) + ξ (E,3) (5) 0.604; Pr (ξ,e,3) ({3, 7}) = ξ (E,3) (3) + ξ (E,3) (7) 0.473; Pr (ξ,e,3) ({5, 7}) = ξ (E,3) (5) + ξ (E,3) (7) ; Pr (ξ,e,3) ({2, 3, 5}) = ξ (E,3) (2) + ξ (E,3) (3) + ξ (E,3) (5) ; Pr (ξ,e,3) ({2, 3, 7}) = ξ (E,3) (2) + ξ (E,3) (3) + ξ (E,3) (7) 0.983; Pr (ξ,e,3) ({2, 5, 7}) = ξ (E,3) (2) + ξ (E,3) (5) + ξ (E,3) (7) ; Pr (ξ,e,3) ({3, 5, 7}) = ξ (E,3) (3) + ξ (E,3) (5) + ξ (E,3) (7) 0.659; Pr (ξ,e,3) ({2, 3, 5, 7}) = ξ (E,3) (2) + ξ (E,3) (3) + ξ (E,3) (5) + ξ (E,3) (7) =. In Example, the value Pr (ξ,e,3) ({2}) is much larger than Pr (ξ,e,3) ({3, 5, 7}). This is a particular case of a general characteristic of the probability functions Pr (ξ,e,v) : from the definitions of ξ (E,v) and Pr (ξ,e,v), one can easily deduce that the values ξ (E,v) (p) decrease fast as p grows.

4 60 G. Sburlati 3. The expected value of ξ (E,v) E and v being fixed, the real number H(ξ (E,v) ) = [p ξ (E,v) (p)] is called expected value of the random variable ξ (E,v). For any E and v we have ( H(ξ (E,v) ) = ln + p (p ) t(v) = [ ( ln + ) = )] [ ( (p ) t(v) + ln( + = + ln [ ( + [( ln + ) ] p (p ) t(v) ) p ] (p ) t(v) ) )] p (p ) t(v) ( ) = + ln ( p )t(v) [(p ) t(v) ] + (p ) t(v). Then + ln ( p )t(v) 2 H(ξ (E,v) ) < + ln ( p )t(v) e, i.e. (p ) t(v) ln 2 () + H(ξ (E,v) ) < + (p ) t(v). In particular (2) H(ξ (E,v) ) < + ζ(t(v) ), ζ denoting Riemann s function. Relation (2) implies that for any choice of E and v the value H(ξ (E,v) ) is finite. EXAMPLE 2. Let us take E = {2, 3, 5, 7} and v = 3. By using appropriate approximations of the values taken by ξ (E,3), we can write H(ξ (E,3) ) = 2 ξ (E,3) (2) +

5 Random variables and congruences 6 3 ξ (E,3) (3)+5 ξ (E,3) (5)+7 ξ (E,3) (7) This result agrees with inequalities 93/44 (), which tell us that H(ξ (E,3) ) > + ln 2 > 2.78 and that ln(405/644) 93/44 H(ξ (E,3) ) < + < In this case, since the primes of E are ln(405/644) small, the value H(ξ (E,3) ) is much closer to the lower bound established by () than to the corresponding upper bound. If E is fixed with 2 E, we can observe that for any v 3 we have L {2} (v) = 2 and, if E = {2}, L E (v) = L {2} (v) L E\{2} (v) = 2 L E\{2} (v) > 2. Therefore, from inequality (2) we deduce (3) H(ξ (E,v) ) < + ζ(t(v) ) ln 2 v N, v 3. For the particular case v = 3 we obtain H(ξ (E,3) ) < + ζ(2) < For generic v, ln 2 if we cannot calculate a sufficiently good approximation of ζ(t(v) ), we can utilise the inequality ζ(r) < r (holding for any r R, r > ), and from inequalities (3) r we derive the weaker relations H(ξ (E,v) ) < + t(v) (t(v) 2) ln 2. Now let us fix v N, v 3 and a generic E with = E P\{2}; let us pose m(e) = min(e). Since L E (v) L {m(e)} (v), from the second of ineqs. () we can obtain (4) H(ξ (E,v) ) < + (p ) t(v) < + + n=m(e) n t(v) ln L {m(e)} (v) < m(e) 2 x t(v) dx ln L {m(e)} (v) = + (t(v) 2) (m(e) 2) t(v) 2 ln L {m(e)} (v). Moreover, we recall the relation L {m(e)} (v) = + and observe that, (m(e) ) t(v) since for any positive x R the inequality ln( + x) > x is verified, replacing x + x by (m(e) ) t(v) gives rise to the relation ln L {m(e)}(v) > (m(e) ) t(v) +. This fact, together with relations (4), implies that H(ξ (E,v) ) < + (m(e) ) t(v) + (t(v) 2)(m(E) 2) t(v) 2.

6 62 G. Sburlati 4. The random variable η (E,v) ( = E P\{2}, v N, v 2) For E and v fixed with = E P\{2} and v N, v 2, we can also define a function η (E,v) : E R related to the values l {p} (v) with p E. Let us set, for each prime p E, η (E,v) (p) = [lnl {p} (v)]/[ln l E (v)]. For any p E, since l E (v) l {p} (v) <, we have ln l E (v) ln l {p} (v) < 0, which implies 0 < η (E,v) (p). Moreover, we can write η (E,v) (p) = lnl {p} (v) lnl E (v) ln l {p} (v) = =. ln l E (v) We have so proven that η (E,v) is a random variable over the set E. A function Pr (η,e,v), defined on the set of all subsets of E, is obtained from η (E,v) by posing I E, Pr (η,e,v) (I) = p I η (E,v) (p); Pr (η,e,v) is easily verified to be a probability over E. EXAMPLE 3. Let us take E = {3, 5, 7, } and v = 2. We have: l {3} (2) = (3 ) 2 = 3 4 ; l {5}(2) = (5 ) 2 = 5 6 ; l {7} (2) = (7 ) 2 = ; l {}(2) = ( ) 2 = ; l E (2) = l {p} (2) = = Such equalities imply that: η (E,2) (3) = ln(3/4) ln(693/024) ; η (E,2)(5) = ln(5/6) ln(693/024) 0.653; η (E,2) (7) = ln(35/36) ln(693/024) ; η (E,2)() = ln(99/00) ln(693/024)

7 Random variables and congruences 63 From all this we deduce that: Pr (η,e,2) ( ) = 0; Pr (η,e,2) ({3}) = η (E,2) (3) ; Pr (η,e,2) ({5}) = η (E,2) (5) 0.653; Pr (η,e,2) ({7}) = η (E,2) (7) ; Pr (η,e,2) ({}) = η (E,2) () ; Pr (η,e,2) ({3, 5}) = η (E,2) (3) + η (E,2) (5) 0.902; Pr (η,e,2) ({3, 7}) = η (E,2) (3) + η (E,2) (7) ; Pr (η,e,2) ({3, }) = η (E,2) (3) + η (E,2) () ; Pr (η,e,2) ({5, 7}) = η (E,2) (5) + η (E,2) (7) ; Pr (η,e,2) ({5, }) = η (E,2) (5) + η (E,2) () 0.90; Pr (η,e,2) ({7, }) = η (E,2) (7) + η (E,2) () ; Pr (η,e,2) ({3, 5, 7}) = η (E,2) (3) + η (E,2) (5) + η (E,2) (7) ; Pr (η,e,2) ({3, 5, }) = η (E,2) (3) + η (E,2) (5) + η (E,2) () ; Pr (η,e,2) ({3, 7, }) = η (E,2) (3) + η (E,2) (7) + η (E,2) () ; Pr (η,e,2) ({5, 7, }) = η (E,2) (5) + η (E,2) (7) + η (E,2) () ; Pr (η,e,2) ({3, 5, 7, }) = η (E,2) (3) + η (E,2) (5) + η (E,2) (7) + η (E,2) () =. Similarly with respect to example, the value Pr (η,e,2) ({3}) found in example 3 is much larger than Pr (η,e,2) ({5, 7, }), and in general the values η (E,v) (p) decrease fast when p grows. 5. The expected value of η (E,v) After fixing E and v, let us consider the expected value of η (E,v), i.e. the real number H(η (E,v) ) = [p η (E,v) (p)]. We have ( H(η (E,v) ) = ln p (p ) s(v) ln l E (v) = [ ( ln ) = )] [ ( (p ) s(v) + ln( = + lnl E (v) [( ln [ ( ) ] ln p (p ) s(v) ln[l E (v) ] ) p ] (p ) s(v) ln l E (v) ) )] p (p ) s(v)

8 64 G. Sburlati Then (5) + = + (p ) s(v) ln[l E (v) ] In particular for v 4 we have ln ( p )s(v) (6) H(η (E,v) ) < + ( (p ) s(v) ln[l E (v) ] ) [(p ) s(v) ]. 4 ln(4/3) (p ) s(v) < H(η (E,v) ) + ln[l E (v). ] 4 ln(4/3) [ζ(s(v) ) ] ln[l E (v). ] Inequality (6) (or, equivalently, the second of ineqs. (5)) proves that for any pair (E, v) with v 4 the value H(η (E,v) ) is finite. For any (E,v) with v 3, from relations (5) it follows that H(η (E,v) ) is finite if and only if the value of the series p is finite. EXAMPLE 4. For E = {3, 5, 7, }, let us calculate the value H(η (E,2) ). We obtain: H(η (E,2) ) = 3 η (E,2) (3)+5 η (E,2) (5)+7 η (E,2) (7)+ η (E,2) () /60 This result agrees with ineqs. (5), which tell us that H(η (E,2) ) > + ln(024/693) > 4 ln(4/3) (6/60) and H(η (E,2) ) + < In this case, the primes of ln(024/693) E are small and hence the value H(η (E,2) ) is closer to the upper bound established by (5) than to the corresponding lower bound. For any pair (E,v) with v 4, starting from the second of ineqs. (5) and using a method like the one at the end of section 3, we can deduce that where m(e) = min(e). H(η (E,v) ) < + 4 ln(4/3) (m(e) )s(v) (s(v) 2)(m(E) 2) s(v) 2, References [] CERRUTI U., Counting the number of solutions of congruences, in: Application of Fibonacci Numbers 5 (Eds. Bergum G.E. et al.), Kluwer Academic Publishers, Dordrecht (The Netherlands) 993, [2] MC CARTHY P.J., Introduction to arithmetical functions, Springer-Verlag, New York 986.

9 Random variables and congruences 65 [3] SBURLATI G., Counting the number of solutions of linear congruences, The Rocky Mountain Journal of Mathematics, 33 (2003), to be published. [4] SBURLATI G., Some properties about counting the solutions of linear congruences, Mathematical Inequalities & Applications, submitted for publication. [5] SIVARAMAKRISHNAN R., Classical theory of arithmetic functions, Marcel Dekker 989. [6] UGRIN-ŠPARAC D., On a Class of Enumeration Problems in Additive Arithmetics, J. Number Theory 45 (993), AMS Subject Classification: D79, 60A99. Giovanni SBURLATI Dipartimento di Fisica Politecnico di Torino corso Duca degli Abruzzi, Torino, ITALIA Present address: Istituto di Informatica e Telematica Area ricerca CNR Via Moruzzi, 5624 Pisa, ITALIA giovanni.sburlati@iit.cnr.it Lavoro pervenuto in redazione il

10 66 G. Sburlati