On The Value Set Of The Ramanujan Function

Arch. Math. 85 (2005) 508–513 0003–889X/05/060508–6 DOI 10.1007/s00013-005-1322-1 © Birkh¨auser Verlag, Basel, 2005

Archiv der Mathematik

On the value set of the Ramanujan function By Igor E. Shparlinski

Abstract. We give lower bounds on the number of distinct values of the Ramanujan function τ (n), n
x, and on the number of distinct residues of τ (n), n
x, modulo a prime . We also show that for any prime  the values τ (n), n
4 , form a finite additive basis modulo .

1. Introduction. Let τ (n) denote the Ramanujan function defined by the expansion X

∞
n=1

(1 − X n )24 =

∞ 

τ (n)X n ,

|X| < 1.

n=1

For a real x  0 we denote by T (x) the value set T (x) = {τ (n) | n
x}. Also for a prime  we denote by T (x; ) the set of residues T (x; ) = {τ (n) (mod ) | n
x}. It is known that if   = 2, 3, 5, 7, 23, 691 and x is sufficiently large then #T (x; ) = , see [7]. An upper bound on the smallest value N for which #T (N ; ) =  can be derived from the Chebotarev Density Theorem, see [5], [8], but it is rather poor, unless one assumes the Extended Riemann Hypothesis. Here we present some simple arguments which yield reasonably strong (taking into account the scarsity of our knowledge about the behaviour of τ (n)) lower bounds for #T (x) and #T (x; ). In particular our lower bound on #T (x; ) is nontrivial for any value of x. Then we combine similar arguments with a recent bound of Bourgain, Katz and Tao [2] to show that for some s the Waring type congruence with the Ramanujan function τ (n1 ) + · · · + τ (ns ) ≡ a (mod ) is solvable for any a with some positive integers n1 , . . . , ns
4 . Mathematics Subject Classification (2000): 11B50, 11F11, 11T23. Partially supported by grant from Lithuanian Foundation of Studies and Science.

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Finally, using a result of Bourgain, Glibichuk and Konyagin [1] we show that the fractional parts {τ (n)/}, n
x, are rather dense in the unit interval. The following basic properties of τ (n) underlie our approach, which is similar to those of [6], [9]: • τ (n) is an integer-valued multiplicative function, that is, τ (m)τ (n) = τ (mn) if gcd(m, n) = 1. • For any prime p, and an integer r  0, τ (p r+2 ) = τ (p r+1 )τ (p) − p11 τ (p r ), where τ (1) = 1. In particular, the identity (1)

τ (p2 ) = τ (p)2 − p 11

plays a crucial role in our arguments. It is also useful to recall that by the famous result of Deligne (2)

|τ (n)|
n11/2+o(1)

see [5]. Throughout the paper, the implied constants in the symbols “O”, “” and “” are absolute (recall that the notations U  V and V  U are equivalent to the statement that U = O(V ) for positive functions U and V ). We also use the symbol “o” with its usual meaning: the statement U = o(V ) is equivalent to U/V → 0. 2. Lower Bounds on #T (x) and #T (x; ) ). Theorem 1. The following bound holds: #T (x)  x 1/3+o(1) . P r o o f. If τ (p) takes more than y values for p
x 1/2 then by the multiplicativity and the bound d(k) = k o(1) on the number d(k) of integer divisors of an integer k  1, see [4, Theorem 317], we derive that τ (p1 p2 ) takes at least y 2 x o(1) values. Otherwise τ (p) = a for some a and at least π(x 1/2 )/y = x 1/2+o(1) y −1 primes p
x 1/2 . Then for all such primes, we see from (1) that the values of τ (p2 ) are all distinct. Taking y = x 1/6 we finish the proof.
The same simple argument, which has been used in the proof of Theorem 1 also works for #T (x; ) however the bound is a little weaker. Theorem 2. The following bound holds: #T (x; )  min{1/2+o(1) , x 1/4+o(1) }.

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I. E. Shparlinski

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P r o o f. Obviously it is enough to prove the result for x
2 . If τ (p) takes more than y residues classes modulo  for p
x 1/2 then of course #T (x; )  y. Otherwise we deduce that τ (p) ≡ a (mod ) for some a and at least π(x 1/2 )/y = 1/2+o(1) x y −1 primes p
x 1/2 . Then for all such primes, by (1), we have τ (p2 ) = τ (p)2 − p 11 ≡ a 2 − p 11 (mod ). Clearly the total number of primes p
x 1/2
 with the same value of p 11 (mod ) is at most 11. Therefore the number of distinct residue classes modulo  taken by τ (p 2 ) is at least x 1/2+o(1) y −1 . Taking y = x 1/4 we finish the proof.
3. Waring Type Congruence for τ (n) (n). Theorem 3. There exists an absolute integer constant s  1 such that the congruence τ (n1 ) + · · · + τ (ns ) ≡ a (mod ) is solvable for any a with some positive integers n1 , . . . , ns
4 . P r o o f. We can assume that  is sufficiently large because otherwise the result is trivial. We consider the sets Sν = {τ (p ν ) | p
, p ≡ 1 (mod 4)},

ν = 1, 2.

Let y > 0 be an arbitrary real number. First of all, assume that #S1  y. By [2], there is a constant α such that either the cardinality of the set of sums of distinct elements of S1 satisfies # {τ (p1 ) + τ (p2 ) | p1 , p2 ∈ S1 , p1  = p2 }  y 1+α , or the cardinality of the set of products of distinct elements of S1 satisfies # {τ (p1 )τ (p2 ) | p1 , p2 ∈ S1 , p1  = p2 }  y 1+α . On the other hand, if #S1 < y, then as in the proof of Theorem 2, we derive that #S2  1+o(1) /y. Taking y = 1/(2+α) we see that least one of the above sets, which we denote by U, is of cardinality at least #U  (1+β)/2+o(1) where β = α/(2 + α). Similarly, using primes p
 with p ≡ 3 (mod 4), we construct a set V, of cardinality at least #V  (1+β)/2+o(1) .

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Let us define e (z) = exp(2π iz/). We now recall the well known bound of double exponential sums        (3) e (λuv)  (#U#V)1/2 max   gcd(λ,)=1  u∈U v∈V

see, for example, Exercise 14.a in Chapter 6 of [10]. For the number Tr (a) of solutions to the congruence u1 v1 + · · · + ur vr ≡ a (mod ), where u1 , . . . , ur ∈ U, v1 , . . . , vr ∈ V, we derive 1 e (λ (u1 v1 + · · · + ur vr − a))  λ=0 u1 ,...,ur ∈U v1 ,...,vr ∈V  r −1  1 e (−λa) e (λuv) =  λ=0 u∈U v∈V r   −1   1     (#U#V)r  +O = e (λuv) .     

Tr (a) =





−1

λ=1 u∈U v∈V

Using (3), we obtain Tr (a) =

(#U#V)r + O((#U#V)r/2 ). 

Because #U#V  1+β+o(1) , we see that for r = 2β −1 + 1 > 2β −1 we have (#U#V)r .  Thus Tr (a) > 0 if  is large enough. Recalling the multiplicativity of τ (n), we see that U and V both consist of the elements either of the shape τ (p1 ) + τ (p2 ), or of the shape τ (p1 )τ (p2 ) = τ (p1 p2 ), or of the shape τ (p2 ) for some primes p, p1 , p2 , and that by our construction the corresponding primes are distinct in U and V. In particular, a product of two elements uv for u ∈ U and v ∈ V is a sum of either of one, or two, or fours values of the τ -function, each with an argument that does not exceed 4 . Therefore we obtain the desired result with either s = r or s = 2r or s = 4r.
Tr (a) = (1 + o(1))

4. Distribution of Fractional Parts {τ (n)/} (n)/}. Theorem 1. There exist absolute constants A > 0 and α > 0 such that for any real η ∈ [0, 1] there exist a positive integer n ≤ A with      τ (n)  − η  −α .  

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P r o o f. We can assume that  is sufficiently large because otherwise the result is trivial. As we have seen in the proof of Theorem 3, for the sets Sν = {τ (p ν ) | p
2 },

ν = 1, 2,

we have max{#S1 , #S2 }  1/2+o(1) . Thus either for ν = 1 or for ν = 2 there exits a set P of t  1/2+o(1) primes p
2 such that S = {τ (pν ) | p ∈ P} is of cardinality #S = #P = t  1/2+o(1) . As before, let e (z) = exp(2π iz/). By Theorem 5 of [1] we see that there is an absolute integer constant k  1 and an absolute real constant γ > 0 such that         e (λs1 · · · sk )  t k −γ . max  gcd(λ,)=1   s1 ,...,sk ∈S Let N = {n = p1ν . . . pkν | p1 < . . . < pk , p1 , . . . , pk ∈ P}. Recalling that τ (n) is multiplicative, we obtain   e (λτ (n)) = e (λτ (p1ν · · · pkν )) = n∈N

p1 <...
=

1 k!





e (λs1 · · · sk )

s1 <...<sk s1 ,... ,sk ∈S

e (λs1 · · · sk ) + O(t k−1 ).

s1 ,...,sk ∈S

Since t  1/2+o(1) and  t #N =  tk, k we obtain (4)

      e (λτ (n))  t k −γ + t k−1  t k −β  #N −β , max   gcd(λ,)=1  n∈N

where β = min{γ , 1/3}. Then the result immediately follows from (4) with any α < β and A = 2k.


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5. Remarks. Taking into account the expected rate of growth of τ (n), (it is expected that the bound (2) reflects the correct order of magnitude of τ (n), see [5]) one can probably safely assume that indeed #T (x) = x 1+o(1) or maybe even #T (x) = (1 + o(1))x. It is also clear that having a good upper bound (of the shape ho(1) ) on the number of solutions of the Diophantine equation u2 = v 11 + h would allow to improve the result of Theorem 1 as #T (x)  x 1/2+o(1) . We remark that according to the famous uniformity conjecture of L. Caporaso, J. Harris and B. Mazur [3], the number of solutions of this equation should be bounded by an absolute constant for all h. Unfortunately, all known bounds on the number of solutions of this and similar equations are of the form hO(1) , which is too weak to be useful. The results of Theorems 3 and 4 can easily be extended to the congruences τ (n1 )m + · · · + τ (nr )m ≡ a (mod ) and to the fractional parts {τ (n)m /}. The bound of exponential sums of [1], which underlies the proof of Theorem 4, can probably be made more explicit, which would immediately lead to concrete numerical values for A and α in the result of Theorem 4. A c k n o w l e d g e m e n t s. The author wishes to thank Florian Luca, Ram Murty and Ren´e Schoof for useful discussions and references. This work was supported in part by ARC grant DP0211459. References [1] J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order. Preprint 2004. [2] J. Bourgain, N. Katz and T. Tao, A sum-product estimate in finite fields and their applications. Geom. Func. Anal. 14, 27–57 (2004). [3] L. Caporaso, J. Harris and B. Mazur, Uniformity of rational points. J. Amer. Math. Soc. 10, 1–35 (1997). [4] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers. Oxford 1979. [5] M. R. Murty, The Ramanujan τ function. Ramanujan Revisited, Proc. Illinois Conference on Ramanujan. 269–288 (1988). [6] M. R. Murty, V. K. Murty and T. N. Shorey, Odd values of the Ramanujan τ -function. Bull. Soc. Math. France. 115, 391–395 (1987). [7] J-P. Serre, Congruences et formes modulaires [d’apr`es H. P. F. Swinnerton-Dyer], LNM 317, 319–338 Berlin-Heidelberg-New York 1973. ´ [8] J-P. Serre, Quelques applications du th´eor`eme de densit´e de Chebotarev. Publ. Math. Inst. Hautes Etud. Sci. 54, 123–201 (1981). [9] T. N. Shorey, Ramanujan and binary recursive sequences. J. Indian Math. Soc. (N.S.), 52, 147–157 (1987). [10] I. M. Vinogradov, Elements of number theory. Dover 1954. Received: 6 October 2004 Igor E. Shparlinski Department of Computing Macquarie University Sydney, NSW 2109 Australia [email protected]