Roma Tre April 8 0, 08 4th Mini Symposium of the Roman umber Theory Association (RTA) Representation of integers by cyclotomic binary forms Michel Waldschmidt Institut de Mathématiques de Jussieu Paris VI http://wwwimj-prgfr/~michelwaldschmidt/ Abstract The homogeneous form n (X, Y ) of degree '(n) which is associated with the cyclotomic polynomial n(t) is dubbed a cyclotomic binary form A positive integer m is said to be representable by a cyclotomic binary form if there exist integers n, x, y with n 3 and max{ x, y } such that n(x, y) =m These definitions give rise to a number of questions that we plan to address update: 3/04/08 /49 /49 This is a joint work with Étienne Fouvry and Claude Levesque Cyclotomic polynomials Definition by induction : (t) =t, t n = Y d n d(t) For p prime, t p =(t )(t p + t p + + t + ) = (t) p (t), Étienne Fouvry Claude Levesque so p(t) =t p + t p + + t + Representation of integers by cyclotomic binary forms Acta Arithmetica, 0p Online First March 08 arxiv: 70909 [matht] For instance (t) =t+, 3(t) =t +t+, 5(t) =t 4 +t 3 +t +t+ 3/49 4/49
Cyclotomic polynomials Cyclotomic polynomials and roots of unity n(t) = tn Y d(t) d6=n d n For n, if is a primitive n th root of unity, n(t) = Y (t j ) gcd(j,n)= For instance 6(t) = The degree of function 4(t) = t4 t = t + = (t ), t 6 (t 3 )(t + ) = t3 + t + = t t + = 3 ( t) n(t) is '(n), where ' is the Euler totient For n, n(t) is the irreducible polynomial over Q of the primitive n th roots of unity, Let K be a field and let n be a positive integer Assume that K has characteristic either 0 or else a prime number p prime to n Then the polynomial n(t) is separable over K and its roots in K are exactly the primitive n th roots of unity which belong to K 5/49 6/49 Properties of For n n(t) we have n(t) =t '(n) n(/t) Let n = e 0 p e p er r where p,,p r are di erent odd primes, e 0 0, e i for i =,,r and r Denote by R the radical of n, namely ( p p r if e 0, R = p p r if e 0 = 0 Then, Let n = m with m odd n(t) = R (t n/r ) 3 Then n(t) = m ( t) n() For n, we have n() =e (n), where the von Mangoldt function is defined for n as ( log p if n = p r with p prime and r ; (n) = 0 otherwise In other terms we have ( p if n = p r with p prime and r ; n() = otherwise 7/49 8/49
n( ) For n 3, n( ) = ( if n is odd ; n/() if n is even Lower bound for n(t) For n 3, the polynomial n(t) has real coe cients and no real root, hence it takes only positive values (and its degree '(n) is even) For n 3 and t R, we have n(t) '(n) In other terms, for n 3, ( p if n = p r with p aprimeandr ; n( ) = otherwise Hence n( ) = when n is odd or when n = m where m has at least two distinct prime divisors Consequence : from n(t) =t '(n) n(/t) we deduce, for n 3 and t R, n(t) '(n) max{, t } '(n) 9/49 0 / 49 n(t) '(n) for n 3 and t R Proof Let n be a primitive n-th root of unity in C ; where n(t) = Q( n)/q(t n )= Y (t ( n )), runs over the embeddings Q( n )! C We have t ( n ) =m( ( n )) > 0, (i)=m( ( n )) = ( n ) ( n )= ( n n ) ow (i)=m( n )= n n Q( n ) is an algebraic integer : '(n) n(t) Q( n)/q((i)=m( n )) The cyclotomic binary forms For n, define n(x, Y )=Y '(n) n(x/y ) This is a binary form in Z[X, Y ] of degree '(n) Consequence of the lower bound for n(t) : for n 3 and (x, y) Z, Therefore, if n(x, y) '(n) max{ x, y } '(n) n(x, y) =m, then max{ x, y } apple m /'(n) If max{ x, y } 3, then n is bounded : '(n) apple log m log(3/) / 49 / 49
Generalization to CM fields (Győry, 977) Kálmán Győry, László Lovász Let K be a CM field of degree d over Q Let K be such that K = Q( ) ; let f be the irreducible polynomial of over Q and let F (X, Y )=Y d f(x/y ) the associated homogeneous binary form : f(t) =a 0 t d + a t d + + a d, F (X, Y )=a 0 X d + a X d Y + + a d Y d For (x, y) Z we have x d apple d a d d F (x, y) and y d apple d a d 0 F (x, y) K Győry L Lovász K Győry & L Lovász, Representation of integers by norm forms II, Publ Math Debrecen 7, 73 8, (970) K Győry, Représentation des nombres entiers par des formes binaires, Publ Math Debrecen 4, 363 375, (977) 3 / 49 4 / 49 Best possible for CM fields Let n 3, not of the form p a nor p a with p prime and a, so that n() = n ( ) = Then the binary form F n (X, Y )= n (X, Y X) has degree d = '(n) and a 0 = a d = Forx Z we have F n (x, x) = n (x, x) =x d Hence, for y = x, we have y d = d a d 0 F (x, y) Binary cyclotomic forms (EF CL MW 08) Let m be a positive integer and let n, x, y be rational integers satisfying n 3, max{ x, y } and n(x, y) = mthen max{ x, y } apple p m /'(n), hence '(n) apple log m 3 log 3 These estimates are optimal, since for `, 3(`, `) =3` If we assume '(n) >, namely '(n) '(n) apple 4 log log m 4, then 5 / 49 which is best possible since 5(, ) = 6 / 49
Lower bound for the cyclotomic polynomials The sequence (c n ) n 3 The upper bound max{ x, y } apple p 3 m /'(n) for n(x, y) =m is equivalent to the following result : For n 3 and t R, n(t) p! '(n) 3 Let n 3 Write c n = inf tr n (t) (n 3) n = e 0 p e p er r where p,,p r are odd primes with p < < p r, e 0 0, e i for i =,,r and r 0 (i) For r = 0, we have e 0 and c n = c e 0 = (ii) For r we have c n = c p p r p r 7 / 49 8 / 49 End of the proof of n(t) p! '(n) 3 The sequence (c n ) n 3 n(x, y) c n max{ x, y } '(n) Lemma For any odd squarefree integer n = p p r with p < p < < p r satisfying n and n 6= 5, we have '(n) > r+ log p p! '(n) 3 c n lim inf c n = 0 and lim sup c n = n! n! The sequence (c p ) p odd prime is decreasing from 3/4 to / For p and p primes, c p p p For any prime p, lim p! c p p = p 9 / 49 0 / 49
The sequence (a m ) m For each integer m, the set (n, x, y) Z n 3, max{ x, y }, n(x, y) =m is finite Let a m the number of its elements The sequence of integers m such that a m starts with the following values of a m m 3 4 5 7 8 9 0 3 6 7 a m 8 6 8 4 4 6 8 8 40 40 6 OEIS A994 https://oeisorg/a994 umber of representations of integers by cyclotomic binary forms The sequence (a m ) m starts with 0, 0, 8, 6, 8, 0, 4, 4, 6, 8, 8,, 40, 0, 0, 40, 6, 4, 4, 8, 4, 0, 0, 0, 4, 8,, 4, 8, 0, 3, 8, 0, 8, 0, 6, 3, 0, 4, 8, 8, 0, 3, 0, 8, 0, 0,, 40,, 0, 3, 8, 0, 8, 0, 3, 8, 0, 0, 48, 0, 4, 40, 6, 0, 4, 8, 0, 0, 0, 4, 48, 8,, 4, / 49 / 49 OEIS A96095 OEIS A93654 https://oeisorg/a96095 Integers represented by cyclotomic binary forms https://oeisorg/a93654 Integers not represented by cyclotomic binary forms a m 6= 0 for m = 3, 4, 5, 7, 8, 9, 0,,, 3, 6, 7, 8, 9, 0,, 5, 6, 7, 8, 9, 3, 3, 34, 36, 37, 39, 40, 4, 43, 45, 48, 49, 50, 5, 53, 55, 57, 58, 6, 63, 64, 65, 67, 68, 7, 73, 74, 75, 76, 79, 80, 8, 8, 84, 85, 89, 90, 9, 93, 97, 98, 00, 0, 03, 04, 06, 08, 09,,, 3, 6, 7,,, a m = 0 for m =,, 6, 4, 5,, 3, 4, 30, 33, 35, 38, 4, 44, 46, 47, 5, 54, 56, 59, 60, 6, 66, 69, 70, 7, 77, 78, 83, 86, 87, 88, 9, 94, 95, 96, 99, 0, 05, 07, 0, 4, 5, 8, 9, 0, 3, 6, 3, 3, 34, 35, 38, 40, 4, 4, 43, 50, 3 / 49 4 / 49
Integers represented by cyclotomic binary forms For, let A() be the number of m apple which are represented by cyclotomic binary forms : A() =#{m m apple, a m 6= 0} = 3 + 4 The number of positive integers apple represented by 4 (namely the sums of two squares) is! 4 + O (log ) (log ) 3 We have A() = (log ) as! (log ) 3 4 + O (log ) 3! The number of positive integers apple represented by 3 (namely x + xy + y : Loeschian numbers) is! 3 + O (log ) (log ) 3 The number of positive integers apple represented by 4 and by 3 is! + O (log ) 3 4 (log ) 7 4 5 / 49 6 / 49 The Landau Ramanujan constant OEIS A064533 OEIS A064533 Decimal expansion of Landau-Ramanujan constant 4 = 0764 3 653 589 0 Edmund Landau 877 938 Srinivasa Ramanujan 887 90 The number of positive integers apple which are sums of two squares is asymptotically 4 (log ) /, where 4 = Y p 3 mod 4 p Ph Flajolet and I Vardi, Zeta function expansions of some classical constants, Feb 8 996 Xavier Gourdon and Pascal Sebah, Constants and records of computation David E G Hare, 5 079 digits of the Landau-Ramanujan constant 7 / 49 8 / 49
The Landau Ramanujan constant References : https://oeisorg/a064533 B C Berndt, Ramanujan s notebook part IV, Springer-Verlag, 994 S R Finch, Mathematical Constants, Cambridge, 003, pp 98-04 G H Hardy, Ramanujan, Twelve lectures on subjects suggested by his life and work, Chelsea, 940 Institute of Physics, Constants - Landau-Ramanujan Constant Simon Plou e, Landau Ramanujan constant Eric Weisstein s World of Mathematics, Ramanujan constant https://enwikipediaorg/wiki/landau-ramanujan_constant Sums of two squares If a and q are two integers, we denote by a,q any integer satisfying the condition An integer m p a,q =) p a mod q is of the form m = 4 (x, y) =x + y if and only if there exist integers a 0, 3,4 and,4 such that m = a 3,4,4 9 / 49 30 / 49 Loeschian numbers : m = x + xy + y An integer m is of the form m = 3 (x, y) = 6 (x, y) =x + xy + y if and only if there exist integers b 0,,3 and,3 such that m = 3 b,3,3 The number of positive integers apple which are represented by the quadratic form X + XY + Y is asymptotically 3 (log ) / where 3 = 3 4 Y p mod 3 p OEIS A3049 OEIS A3049 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers 3 = 3 4 Y p mod 3 3 = 0638 909 405 44 p = 3 + 4 = 403 33 059 3 / 49 3 / 49
Zeta function expansions of some classical constants, Feb 8 996 OEIS A30430 OEIS A30430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares Philippe Flajolet Ilan Vardi 3 = 063890940544534388 549467498450937 549755080933454 69365708076300764 96584689779586 643884687590744 = 3 4 p 5 (log(+ 3)) 4 4 (/4) Y p 5, 7, mod = 030 36 4 35 Only digits after the decimal point are known p Bill Allombert 33 / 49 34 / 49 Zeta function expansions of some classical constants, Feb 8 996 Further developments Philippe Flajolet Bill Allombert Ilan Vardi = 030364357065637 947769900480997560 4795893696454588 6784888654487540 50899487467839797 708567765937590 Prove similar estimates for the number of integers represented by other binary forms (done for quadratic forms) ; eg : prove similar estimates for the number of integers which are sums of two cubes, two biquadrates, Prove similar estimates for the number of integers which are represented by n for a given n Prove similar estimates for the number of integers which are represented by n for some n with '(n) d 35 / 49 36 / 49
Stewart - Xiao Cam Stewart and Stanley Yao Xiao Let F be a binary form of degree d 3 with nonzero discriminant There exists a positive constant C F > 0 such that the number of integers of absolute value at most which are represented by F (X, Y ) is asymptotic to C F /d Cam Stewart Stanley Yao Xiao CL Stewart and S Yao Xiao, On the representation of integers by binary forms, arxiv:6050347v (March 3, 08) 37 / 49 38 / 49 K Mahler (933) Kurt Mahler Let F be a binary form of degree d 3 with nonzero discriminant Denote by A F the area (Lebesgue measure) of the domain {(x, y) R F (x, y) apple } For Z > 0 denote by F (Z) the number of (x, y) Z such that 0 < F (x, y) apple Z Then F (Z) =A F Z /d + O(Z /(d ) ) as Z! Kurt Mahler 903 988 Über die mittlere Anzahl der Darstellungen grosser Zahlen durch binäre Formen, Acta Math 6 (933), 9-66 https://carmanewcastleeduau/mahler/biographyhtml 39 / 49 40 / 49
Higher degree The situation for positive definite forms of degree 3 is di erent for the following reason : If a positive integer m is represented by a positive definite quadratic form, it usually has many such representations ; while if a positive integer m is represented by a positive definite binary form of degree d 3, it usually has few such representations If F is a positive definite quadratic form, the number of (x, y) with F (x, y) apple is asymptotically a constant times, but the number of F (x, y) is much smaller If F is a positive definite binary form of degree d 3, the number of (x, y) with F (x, y) apple is asymptotically a constant times /d, the number of F (x, y) is also asymptotically a constant times /d Sums of k th powers If a positive integer m is a sum of two squares, there are many such representations Indeed, the number of (x, y) in Z Z with x + y apple is asymptotic to, while the number of values apple taken by the quadratic form 4 is asymptotic to 4 / p log where 4 is the Landau Ramanujan constant Hence 4 takes each of these values with a high multiplicity, on the average ( / ) p log On the opposite, it is extremely rare that a positive integer is a sum of two biquadrates in more than one way (not counting symmetries) 4 / 49 4 / 49 635 38 657 = 58 4 + 59 4 = 34 4 + 33 4 Sums of k th powers Leonhard Euler 707 783 The smallest integer represented by x 4 + y 4 in two essentially di erent ways was found by Euler, it is 63538657 = 4 3 4 569 One conjectures that given k 5, if an integer is of the form x k + y k, there is essentially a unique such representation But there is no value of k for which this has been proved [OEIS A684] umber of solutions to the equation x 4 + y 4 = n with x y > 0 An infinite family with one parameter is known for non trivial solutions to x 4 + x 4 = x 4 3 + x 4 4 http://mathworldwolframcom/diophantineequation4thpowershtml 43 / 49 44 / 49
Higher degree The situation for positive definite forms of degree di erent also for the following reason 3 is Quartan primes [OEIS A00645] Quartan primes: primes of the form x 4 + y 4, x > 0, y > 0 A necessary and su cient condition for a number m to be represented by one of the quadratic forms 3, 4, is given by acongruence By contrast, consider the quartic binary form 8(X, Y )=X 4 + Y 4 On the one hand, an integer represented by 8 is of the form,8 ( 3,8 5,8 7,8 ) 4 On the other hand, there are many integers of this form which are not represented by 8 The list of prime numbers represented by 8 start with, 7, 97, 57, 337, 64, 88, 97, 47, 657, 3697, 477, 47, 6577, 0657, 40, 4657, 4897, 5937, 656, 887, 3856, 3904, 4997, 547, 65537, 6567, 666, 66977, 8077, 83537, 83777, 8904, 0560, 07377, 967, It is not known whether this list is finite or not The largest known quartan prime is currently the largest known generalized Fermat prime: The 35365-digit (4530 65536 ) 4 + 4 45 / 49 46 / 49 Primes of the form x k + y k Primes of the form X + Y 4 [OEIS A0033] primes of the form x + y [OEIS A00645] primes of the form x 4 + y 4, [OEIS A006686] primes of the form x 8 + y 8, [OEIS A0066] primes of the form x 6 + y 6, [OEIS A0067] primes of the form x 3 + y 3 John Friedlander Étienne Fouvry But it is known that there are infinitely many prime numbers of the form X + Y 4 Friedlander, J & Iwaniec, H The polynomial X + Y 4 captures its primes, Ann of Math () 48 (998), no 3, 945 040 https://arxivorg/pdf/math/9885pdf 47 / 49 48 / 49