¨ AND CERTAIN DIRICHLET SERIES ON A CONJECTURE OF ERDOS
arXiv:1501.04185v1 [math.NT] 17 Jan 2015
TAPAS CHATTERJEE1 AND M. RAM MURTY2
A BSTRACT. Let f : Z/qZ → Z be such that f (a) = ±1 for 1 ≤ a < q, and f (q) = 0. Then Erdos ¨ P = 6 0. For q even, it is easy to show that the conjecture is true. The case conjectured that n≥1 f (n) n q ≡ 3 ( mod 4) was solved by Murty and Saradha. In this paper, we show that this conjecture is true for 82% of the remaining integers q ≡ 1 ( mod 4).
1. Introduction In a written communication with Livingston, Erdos ¨ made the following conjecture (see [5] ): if f is a periodic arithmetic function with period q and ( ±1 if q ∤ n, f (n) = 0 otherwise, then L(1, f ) =
∞ X f (n)
n=1
n
6= 0
where the L-function L(s, f ) associated with f is defined by the series L(s, f ) :=
∞ X f (n)
n=1
ns
.
(1)
In 1973, Baker, Birch and Wirsing (see Theorem 1 of [1]), using Baker’s theory of linear forms in logarithms, proved the conjecture for q prime. In 1982, Okada [10] established the conjecture if 2ϕ(q) + 1 > q. Hence, if q is a prime power or a product of two distinct odd primes, the conjecture is true. In 2002, R. Tijdeman [12] proved the conjecture is true for periodic completely multiplicative functions f . Saradha and Tijdeman [11] showed that if f is periodic and multiplicative with |f (pk )| < p − 1 for every prime divisor p of q and every positive integer k, then the conjecture is true. It is easy to see that ∞ X f (n) L(1, f ) = n n=1 Pq exists if and only if n=1 f (n) = 0. If q is even and f takes values ±1 with f (q) = 0, then P q n=1 f (n) 6= 0. Hence the conjecture holds for even q. In 2007, Murty and Saradha [8] proved that if q is odd and f is an odd integer valued odd periodic function then the conclusion of the conjecture holds. In 2010, they proved that the 2010 Mathematics Subject Classification. 11M06, 11M20. Key words and phrases. Erdos ¨ conjecture, non-vanishing of Dirichlet series, Okada’s criterion. 1 Research of the first author was supported by a post doctoral fellowship at Queen’s University. 2 Research of the second author was supported by an NSERC Discovery grant and a Simons fellowship. 1
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TAPAS CHATTERJEE AND M. RAM MURTY
Erdos ¨ conjecture is true if q ≡ 3 ( mod 4) (see Theorem 7 of [7]). Thus the conjecture is open only in cases where q ≡ 1 (mod 4). However, it seems that a novel idea will be needed to deal with these cases. In this paper, we adopt a new density-theoretic approach which is orthogonal to earlier methods. Here is the main consequence of our method:
¨ conjecture is true for q}|. Theorem 1.1. Let S(X) = |{q ≡ 1 ( mod 4), q ≤ X| Erdos Then S(X) ≥ 0.82. lim inf X→∞ X/4 In other words, the Erdos ¨ conjecture is true for at least 82 % of the integers q ≡ 1 (mod 4). Our method does not extend to show that the Erdos ¨ conjecture is true for 100 % of the moduli q ≡ 1 (mod 4). We examine this question briefly at the end of the paper. It seems to us that more ideas are needed to resolve the conjecture fully. These questions have a long history beginning with Baker, Birch and Wirsing [1]. Their work was generalized by Gun, Murty and Rath [4] to the setting of algebraic number fields. The forthcoming paper [2] gives new proofs of some of the background results of this area. We also refer the reader to [12] for an expanded survey of the early history. 2. Notations and Preliminaries From now onwards, we denote the field of rationals by Q, the field of algebraic numbers by Q, Euler’s totient function by ϕ and Euler’s constant by γ. We say a function f is Erdosian ¨ mod q if f is a periodic function with period q and ( ±1 if q ∤ n, f (n) = 0 otherwise. Also we will write f (X) . g(X) to mean lim sup X→∞
f (X) ≤ 1. g(X)
Similarly, we write f (x) & g(x) to mean lim inf X→∞
f (X) ≥ 1. g(X)
2.1. Okada’s criterion. Proposition 2.1. Let the q-th cyclotomic polynomial Φq be irreducible over the field Q(f (1), · · · , f (q)). Let M (q) be the set of positive integers which are composed of prime factors of q. Then L(1, f ) = 0 if and only if X
m∈M (q)
for every a with 1 ≤ a < q, (a, q) = 1, and
q X
r=1 (r,q)>1
f (am) =0 m
f (r)ǫ(r, p) = 0
¨ ON A CONJECTURE OF ERDOS
3
for every prime divisor p of q, where ( vp (r) ǫ(r, p) = vp (q) +
1 p−1
if vp (r) < vp (q), otherwise
and for any integer r, vp (r) is the exponent of p dividing r. This Proposition is a modification, due to Saradha and Tijdeman [11], of a 1986 result of Okada [9]. Note that Okada deduced the sufficient condition 2ϕ(q) + 1 > q stated in the introduction from his original version of this criterion. 2.2. Wirsing’s Theorem. The following proposition is due to Wirsing [13]. Proposition 2.2. Let f be a non-negative multiplicative arithmetic function, satisfying |f (p)| ≤ G for all primes p, X
p−1 f (p) log p ∼ τ log X,
p≤X
with some constants G > 0, τ > 0 and XX
p−k |f (pk )| < ∞;
p k≥2
if 0 < τ ≤ 1, then, in addition, the condition X X p
|f (pk )| = O(X/ log X)
k≥2 pk ≤X
is assumed to hold. Then X
f (n) = (1 + o(1))
n≤X
X e−γτ Y f (p) f (p2 ) (1 + + + · · · ). log X Γ(τ ) p p2 p≤X
2.3. Mertens Theorem. We also need a classical theorem of Mertens in a later section. We record the theorem here (see for example, p. 130 of [6]): Q (1 − p1 ) = e−γ . Proposition 2.3. lim log X X→∞
p≤X
3. Exceptions to the conjecture of Erdos ¨ We say that the Erdos ¨ conjecture is false (mod q), if there is an Erdosian ¨ function f for which L(1, f ) = 0. The following proposition plays a fundamental role in our approach. Proposition 3.1. If the Erdos ¨ conjecture is false (mod q ) with q odd, then X 1 1≤ . ϕ(d) d|q d≥3
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TAPAS CHATTERJEE AND M. RAM MURTY
Proof. By the hypothesis, there is an Erdosian ¨ function f mod q for which, we have L(1, f ) = 0. Applying Okada’s criterion, we get X f (b) = 0. b
(2)
b∈M (q)
Let d = (b, q), so that b = db1 with (b1 , q/d) = 1. Then (2) can be written as X1
−f (1) =
d|q d≥3
d
X
b1 ∈M (q) (b1 ,q/d)=1
f (db1 ) . b1
Taking absolute value of both sides, we get 1≤
X1 d|q d≥3
d
X
b1 ∈M (d)
1 . b1
(3)
Notice that the inner sum can be written as X
b1 ∈M (d)
Hence from (3), we get
Y Y 1 d 1 1 1 −1 = = 1 + + 2 + ··· = 1− . b1 p p p ϕ(d) p|d
p|d
1≤
X 1 . ϕ(d) d|q d≥3
Two immediate corollaries of the above proposition are the following:
¨ Corollary 3.2. If q is a prime power or a product of two distinct odd primes, then the Erdos conjecture is true (mod q ). Proof. This is a pleasant elementary exercise.
Hence we have recovered the two basic cases of the conjecture which were given in the introduction, of course, also as a consequence of Okada’s Criterion. Let d(n) be the divisor function, i.e. d(n) is the number of divisors of n. Corollary 3.3. If the smallest prime factor of q is at least d(q), then the Erdos ¨ conjecture is true for q . Proof. Let l be the smallest prime factor of q. From the above proposition, if the Erdos ¨ conjecture is false (mod q), then we have X 1 1 ≤ ϕ(d) d|q d≥3
0, Proposition 3.1 can be rewritten as: Proposition 3.6. If Erd¨os conjecture is false for odd q, then α 1 X 1 1≤ α . 2 ϕ(d) d|q
As before, S1 (X) = |{q ≡ 1 ( mod 4), q ≤ X| Erdos ¨ conjecture is false for q}|. Then from the above proposition, we get α X X 1 1 . S1 (X) ≤ α 2 ϕ(d) q≤X q≡1 ( mod 4)
P Let fα (q) = d|q inequality becomes
1 ϕ(d)
α
d|q
and χ be the non-trivial Dirichlet character mod 4. Then the above
S1 (X) ≤
1 2α+1
X X . χ(q)f (q) f (q) + α α q≤X q odd
q≤X q odd
(4)
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TAPAS CHATTERJEE AND M. RAM MURTY
Again, note that fα (q) is a multiplicative arithmetic function. One can check that it also satisfies all the other hypotheses of Wirsing’s theorem (Proposition 2.2) with G = 2α and τ = 1. So in light of Wirsing’s theorem, we get X fα (p) fα (p2 ) e−γ Y 1+ + + ··· fα (q) ∼ X log X p p2 p≤X p6=2
q≤X q odd
and X
q≤X q odd
e−γ Y χ(p)fα (p) χ(p2 )fα (p2 ) χ(q)fα (q) ∼ X + + ··· . 1+ log X p p2 p≤X p6=2
Again, from Mertens theorem we know that Y e−γ . (1 − 1/p) ∼ log X p≤X
Hence we have
X
fα (q) ∼
q≤X q odd
X Y fα (p) fα (p2 ) (1 − 1/p) 1 + + + ··· 2 p p2 p≤X p6=2
∼
X p1 (say) 2
and X
χ(q)fα (q) ∼
q≤X q odd
X Y χ(p)fα (p) χ(p2 )fα (p2 ) (1 − 1/p) 1 + + + ··· 2 p p2 p≤X p6=2
X p2 (say). 2 Now using the above two inequality (4) becomes, ∼
S1 (X) .
X 2α+2
(p1 + p2 ).
Finally using Maple (see [3]), we find that the quantity on the right hand side is minimized at α ∼ 8.11 and we get X S1 (X) . 0.20 . 4 Hence, we get S(X) ≥ 0.80. lim inf X→∞ X/4 Remarks. One cannot hope to obtain 100 % by these methods. In fact, one can show that there is a positive density (albeit small) of q for which the inequality of Proposition 3.1 holds. Indeed, since X 1 Y 1 ≥ 1+ ϕ(d) p−1 d|q
p|q
we can make the product ( and hence the sum) arbitrarily large by ensuring that q is divisible by all the primes in an initial segment. We can even ensure that these primes are congruent to
¨ ON A CONJECTURE OF ERDOS
9
1 (mod 4). We then take numbers which are divisible by this q and congruent to 1 (mod 4) and deduce that for all these numbers, the inequality in the proposition holds. Since the product on the right diverges slowly to infinity as we go through such numbers q, we obtain in this way, a small density of numbers for which the inequality holds. Acknowledgements. We thank Michael Roth for his help on using the Maple language as well as Sanoli Gun, Purusottam Rath and Ekata Saha for their comments on an earlier version of this paper. We also thank the referee for helpful comments that improved the quality of the paper. R EFERENCES [1] A. Baker, B. J. Birch and E. A. Wirsing, On a problem of Chowla, J. Number Theory 5 (1973), 224-236. [2] T. Chatterjee and M. R. Murty, Non-vanishing of Dirichlet series with periodic coefficients, J. Number Theory, 145 (2014), 1–21. [3] T. Chatterjee and M.R. Murty, Maple code at www.mast.queensu.ca/ murty/maplecode.pdf. [4] S. Gun, M.R. Murty and P. Rath, Linear independence of Hurwitz zeta values and a theorem of Baker-Birch-Wirsing over number fields, Acta Arithmetica, 155 (2012), no. 3, 297–309. P f (n) [5] A. Livingston, The series ∞ n=1 n for periodic f , Canad. Math. Bull. 8 (1965) 413–432. [6] M. Ram Murty, Problems in Analytic Number Theory, 2nd edition, Springer, 2008. [7] M. Ram Murty and N. Saradha, Euler-Lehmer constants and a conjecture of Erd¨os, J. Number Theory 130 (2010), 2671-2682. [8] M. Ram Murty and N. Saradha, Transcendental values of the digamma function, J. Number Theory 125 (2007) 298–318. [9] T. Okada, Dirichlet series with periodic algebraic coefficients, J. London Math. Soc. (2) 33 (1986), no. 1, 13-21. [10] T. Okada, On a certain infinite series for a periodic arithmetical function, Acta Arith. 40 (1982) 143–153. [11] N. Saradha and R. Tijdeman, On the transcendence of infinite sums of values of rational functions, J. London Math. Soc. (3) 67 (2003), 580-592. [12] R. Tijdeman, Some applications of Diophantine approximation, Number Theory for the Millenium III, MA, 2002, pp. 261–284. [13] E. Wirsing, Das asymptotische Verhalten von Summen uber ¨ multiplikative Funktionen, Math. Ann. 143 (1961) 75-102. (T. Chatterjee) D EPARTMENT OF M ATHEMATICS , I NDIAN I NSTITUTE OF T ECHNOLOGY R OPAR , P UNJAB -140001, I NDIA . (M. Ram Murty) D EPARTMENT OF M ATHEMATICS AND S TATISTICS , Q UEEN ’ S U NIVERSITY, K INGSTON , O N C ANADA , K7L3N6. E-mail address, Tapas Chatterjee:
[email protected] E-mail address, M. Ram Murty:
[email protected]
TARIO ,
