ON KUMMER’S CONJECTURE.

M. RAM MURTY AND YIANNIS N. PETRIDIS

Abstract. Kummer conjectured the asymptotic behavior of the first factor of the class number of a cyclotomic field. If we only ask for upper and lower bounds of the order of growth predicted by Kummer, then this modified Kummer conjecture is true for almost all primes.

1. Introduction

Let p be an odd prime and h_{p} be the class number of the cyclotomic field Q(ζ_{p}), where ζ_{p} denotes a primitive p-root of unity. Let h_{p}^{+} denote the class number of the maximal real subfield Q(ζ_{p} + ζ_{p}^{−1}). It is well-known, e.g. [4, p. 213, VI 1.20] that h_{p}^{+}|h_{p} and so we can factor h_{p} = h_{p}^{+}h_{p}^{−}. The number h_{p}^{−} is called the first factor of the class number of the cyclotomic field. In 1851, Kummer [12, p. 473] conjectured that

h_{p}^{−} ∼ \frac{p^{(p+3)/4}}{2^{(p−3)/2}π^{(p−1)/2}} = G(p) (1.1)

as p → ∞. In fact he claimed to have a proof that he would publish later together with further developments. In 1949 Ankeny and Chowla [2], [3] proved that

log(h_{p}^{−}/G(p)) = o(log p) (1.2)

as p → ∞. Unaware of the work of Ankeny and Chowla, Siegel [15] proved the weaker assertion

(1.3) log(h_{p}^{−}/G(p)) = O(p log log p).

He also cast doubt on the original Kummer conjecture and asked whether h_{p}^{−}/G(p) is bounded above and below by positive constants. It follows from (1.2), (1.3) that

log h_{p}^{−} ∼ \frac{1}{4}p log p (1.4)

and that Q(ζ_{p}) has class number 1 for only a finite set of primes p. In 1990 Granville [7] showed that assuming two standard conjectures of analytic number theory, namely the Elliott-Halberstam conjecture and the Hardy-Littlewood conjecture, Kummer’s conjecture is false. More precisely, let π(x, k, l) denote the number of primes p ≤ x with p ≡ l(mod k). The Elliott-Halberstam conjecture predicts that for any ϵ > 0 and A > 0 we have

1991 Mathematics Subject Classification. Primary 11R18; Secondary 11M20.

The first author was partially supported by a Killam Research Fellowship and the Bankers Trust Company Foundation by a grant to the Institute for Advanced Study.

(1.5)  ∑_{k<x^{1−ϵ}} max_{(l,k)=1} max_{y≤x} |π(y, k, l) − \frac{liy}{φ(k)}| ≪_{ϵ,A} \frac{x}{log^{A} x}.

The Hardy-Littlewood conjecture predicts that there are at least ≫ x/ log^{2} x primes p ≤ x such that 2p + 1 is also prime. Assuming these two unproved conjectures, Granville shows that Kummer’s conjecture (1.1) is false.

In this paper we prove the following theorem, which we call the weak Kummer conjecture.

Theorem 1.1. There exists a positive constant c such that for almost all primes

c^{−1} ≤ \frac{h_{p}^{−}}{G(p)} ≤ c. (1.6)

That is, there is a sequence of primes p_{i} such that (1.6) holds and the number of primes p_{i} ≤ x is asymptotic to x/ log x as x → ∞.

This theorem should be compared with further results of Granville in [7, Theorem 5, p. 325] that state that

1/c ≤ h_{p}^{−}/G(p) ≤ c

for a positive proportion ρ(c) of primes p ≤ x, where ρ(c) → 1 as c → ∞. The estimate

log(h_{p}^{−}/G(p)) = O(log log p)

holds for all primes p ∉ S, where S = ∅ under the Generalized Riemann Hypothesis (GRH), or S has density zero unconditionally.

Granville [7] conjectures that

(−\frac{1}{2} + o(1)) log log log p ≤ log(h_{p}^{−}/G(p)) ≤ (\frac{1}{2} + o(1)) log log log p.

In fact he conjectures that they are the best possible bounds, in the sense that there are sequences of primes realizing the left and right equality.

The result in Theorem 1.1 is unconditional. If we assume the Elliott-Halberstam conjecture more can be deduced.

Theorem 1.2. Assume the Elliott-Halberstam conjecture (1.5). There is a constant c > 0 such that for every δ > 0 and for all primes p with the exception of a set P (δ) of zero density in the set of all primes we have

c^{−δ} < \frac{h_{p}^{−}}{G(p)} < c^{δ}

as p → ∞.

Remark 1.3. There are a lot of computations related to Kummer’s conjecture. Kummer himself [12] computed h_{p}^{−} for p < 100. M. Newman [13] extended the calculations to p < 200, after earlier efforts of G. Schrutka von Rechtenstamm [14] for p < 256. D. Lehmer and J. Masley [11] worked out the range 200 < p < 521 and G. Fung, A, Granville and H. Williams [5] computed h_{p}^{−} for p < 3000. The best algorithm known is due to V. Jha [10].

Acknowledgment. The authors would like to thank the referee for helpful comments concerning Theorem 1.2.

2. Preliminary material

We first derive several reformulations and simplifications of Kummer’s conjecture. Much of this material can be found in [3], [7]. Hasse showed that

h_{p}^{−} = G(p) ∏_{χmodp,χ(−1)=−1} L(1, χ), (2.1)

see [4, Theorem 69, 5.13]. This formula follows from the analytic class number formula

∏_{χ∈Θ_{K},χ≠1} L(1, χ) = \frac{2^{s+t}π^{t}R_{K}h_{K}}{W_{K} \sqrt{|d_{k}|}}

applied to Q(ζ_{p}) and Q(ζ_{p} + ζ_{p}^{−1}) and the comparison of the regulators, discriminants and the roots of unity in these fields. Here Θ_{K} is the character group of the (abelian) Galois group of K, R_{K} is the regulator, W_{K} is the number of roots of unity in K, d_{K} is the discriminant and s, t are the number of real and complex imbeddings of K.

Thus

h_{p}^{−} = G(p) exp (\frac{p − 1}{2}f_{p}) , (2.2)

where

f_{p} = lim_{x→∞} f_{p}(x), f_{p}(x) = ∑_{n≤x} \frac{c(n)}{n} (2.3)

with c(n) = 0, unless n is a prime power q^{m}, in which case c(q^{m}) = 1/m if q^{m} ≡ 1(mod p), and −1/m if q^{m} ≡ −1(mod p). Thus Kummer’s conjecture is equivalent to the assertion f_{p} = o(1/p). The weak Kummer conjecture (1.6) is equivalent to f_{p} = O(1/p). As in [7], the Siegel-Walfisz theorem allows to write f_{p} = f_{p}(2^{p}) + o(1/p), for odd primes p. Thus, we may restrict our attention to the finite sum f_{p}(2^{p}). We will use the following results.

Lemma 2.1 (Brun-Titchmarsh Theorem). For k < x, (k, l) = 1 we have

π(x, k, l) < \frac{3x}{φ(k) log(x/k)}. (2.4)

Proof. See [8, Th. 3.8, p. 110].

Lemma 2.2. [9, Hooley, p. 124] Let l be a fixed nonzero integer. Let also ϵ, A and B be any positive real numbers where A ≥ B + 30. Then for any numbers x and X such that x^{1/2} ≤ X < x log^{−A} x and x > x_{0}(ϵ, B), we have

π(x, k, l) ≤ \frac{(4 + ϵ)x}{φ(k) log X} (2.5)

for all values of k satisfying X ≤ k ≤ 2X and (l, k) = 1, except for at most X log^{−B} x exceptional values of k.

Lemma 2.3. Fix l, k, (l, k) = 1. The number of primes x < p ≤ 2x such that kp + l is also prime is

≪ ∏_{p|kl} (1 − \frac{1}{p})^{−1} \frac{x}{log^{2} x}.

Proof. See [8, 2.4.1, p. 80].

Lemma 2.4. There is a constant c such that, as T → ∞,

∑_{k≤T} ∏_{p|k} (1 − \frac{1}{p})^{−1} ∼ cT. (2.6)

Proof. We have ∏_{p|k}(1 − p^{−1}) = φ(k)/k. To analyze the asymptotics of k/φ(k) we look at the Dirichlet series

g(s) = ∑_{k=1}^{∞} \frac{k}{φ(k)} \frac{1}{k^{s}} = ∏_{p} (1 + \frac{p}{(p − 1)(p^{s} − 1)})

and apply the Wiener-Ikehara Tauberian theorem. We only need to notice that g(s)/ζ(s) is regular for ℜs > 1.

Remark 2.5. Siegel’s method [15] is based on the inequality L(1, χ) < 2 log p for the upper bound and two inequalities for the lower bound: Landau’s theorem |L(1, χ)| > c(log p)^{−5} for nonreal characters and the analytic class number formula for a quadratic field

L(1, χ) = \frac{2πh_{Q(\sqrt{−p})}}{W \sqrt{p}} > p^{−1/2}, p ≡ 3(mod 4).

Siegel’s theorem for real characters L(1, χ) > c(ϵ)p^{−ϵ} does not lead to an improvement. Siegel’s doubts on whether Kummer had actually proved his conjecture are based on the fact that neither Kummer nor Dirichlet were aware of Siegel’s theorem.

3. The weak Kummer conjecture: Proof of Theorem 1.1

We need to prove that for all but o(x/ log x) primes p ≤ x we have (1.6). By the remarks of the preceding section, it suffices to consider f_{p}(2^{p}). Moreover, in the definition of f_{p}(x), we can confine our attention to primes as the contribution of prime powers q^{m} with m ≥ 2 is seen to be O(1/p), see [3, p. 490]. Set

(3.1) E(x, p) = max_{y≤x} max_{(a,p)=1} |π(y, p, a) − \frac{lix}{p − 1}| .

The Bombieri-Vinogradov theorem [1] gives for any A > 0 a B = B(A) such that

∑_{p<x^{1/2}/ log^{B} x} E(x, p) ≪_{A} \frac{x}{log^{A} x}. (3.2)

We set

S(t, x) = ∑_{x<p≤2x} E(t, p)

and we can estimate

∑_{x<p≤2x} ∑_{p^{2} log^{2B} p<q≤2^{p}} \frac{c(q)}{q} ≪ [\frac{S(t, x)}{t}]_{x^{2} log^{2B} x}^{∞} + ∫_{x^{2} log^{2B} x}^{∞} \frac{S(t, x)}{t^{2}} dt ≪ log^{−A+1} x. (3.3)

Therefore, if we write f_{p}(2^{p}) = f_{p}(p^{2} log^{2B} p) + D(p), we see that the number of primes p, x < p ≤ 2x, such that |D(p)| > c/p is O(x/ log^{A−1} x). Using a dyadic decomposition of the range p ≤ x into intervals [2^{−i−1}x, 2^{−i}x], we exclude at most x log^{−A+1} x primes p, with p ≤ x and we choose A > 2 in the Bombieri-Vinogradov theorem.

Thus, we may restrict our attention to f_{p}(p^{2} log^{2B} p). We use Lemma 2.1 to observe that

∑_{p^{2}/4<q<p^{2} log^{2B} p} \frac{c(q)}{q} ≪ ∫_{p^{2}/4}^{p^{2} log^{2B} p} \frac{dt}{pt log(t/p)} = o(1/p),

so that we may restrict our attention to f_{p}(p^{2}/4).

We apply Lemma 2.2 in the range [2^{A}p(log p)^{A}, p^{2}/4]. We take k = p in the Lemma and we easily see that, if X ≤ p ≤ 2X, this range is included in the range x^{1/2} ≤ X ≤ x log^{−A} x required in the Lemma. It follows that π(x, p, ±1) ≪ x/(p log p), except for a set of primes p of size at most X log^{−B} X. Again using a dyadic decomposition we exclude at most X log^{−B} X primes p, with p ≤ X. We take B > 2 in Lemma 2.2 and get

∑_{2^{A}p log^{A} p<q<p^{2}/4} \frac{c(q)}{q} ≪ ∫_{2^{A}p log^{A} p}^{p^{2}/4} \frac{x}{x^{2}p log p} dx ≪ \frac{1}{p}

for all but O(X/ log^{B} X) primes p with p ≤ X.

Finally, by Lemma 2.3, the number of primes x < p ≤ 2x such that pk ± 1 is also prime is

≪ ∏_{p|k} (1 − \frac{1}{p})^{−1} \frac{x}{log^{2} x}

for a fixed k. Let the number of primes p ∈ (x, 2x] such that pk ± 1 is also prime for k ≤ ϵ log x/ log log x be N(x). Then

N(x) ≪ ∑_{k≤ϵ log x/ log log x} ∏_{p|k} (1 − \frac{1}{p})^{−1} \frac{x}{log^{2} x} ≪ \frac{ϵx}{log x log log x},

in which we use the estimate (2.6). So we are excluding at most x/(log x log log x) primes p ≤ x. We may therefore suppose that in our summation we have q > ϵp log p/ log log p. Thus we need only to consider

∑_{ϵp log p/ log log p<q<2^{A}p log^{A} p} \frac{c(q)}{q} ≪ ∫_{ϵp log p/ log log p}^{2^{A}p log^{A} p} \frac{dt}{tp log(t/p)}
(3.4)  ≪ \frac{1}{p} log (\frac{log(2^{A} log^{A} p)}{log(ϵ log p/ log log p)}) = O(1/p)

using Lemma 2.1 again. This completes the proof of Theorem 1.1.

Remark 3.1. We see that the exceptional set of primes p ≤ x is ≪ x/ log x log log x.

4. Proof of Theorem 1.2

Proposition 1 in [7] shows that the contribution for m > 1 is o(1/p) with the exception of a set of primes p ≤ x which are at most ≪ x^{1/2} log^{2} x. As in Corollary 1 in [7] the Elliott-Halberstam conjecture (1.5) implies f_{p} = f_{p}(p^{1+δ}) + o(1/p) for all but O_{δ}(x/ log^{3} x) primes p ≤ x. We prove that for almost all primes p we have

(4.1)  ∑_{ϵp log p/ log log p<q<2^{A}p log^{A} p} \frac{c(q)}{q} = o(1/p).

We set q = kp ± 1 and see that, when q varies in the range ϵp log p/ log log p < q < 2^{A}p log^{A} p and x < p ≤ 2x, k varies in the range (ϵ/2) log x/ log log x < k < 2^{A+1} log^{A}(2x). We have

∑_{x<p≤2x} ∑_{q≡±1modp} \frac{1}{q} ≪ ∑_{k} ∑_{x<p≤2x,q=kp±1} \frac{1}{kp}.

We use Lemma 2.3 to estimate the number of primes p between x and 2x with kp±1 also prime by ∏_{p|k}(1 − p^{−1})^{−1}x/ log^{2} x. Using a summation by parts together with (2.6) we easily get that

∑_{k≤T} \frac{1}{k} ∏_{p|k} (1 − \frac{1}{p})^{−1} ≪ log T.

The estimates above give

∑_{x<p≤2x} ∑_{q≡±1modp} \frac{1}{q} ≪ ∑_{k≪log^{A}(2x)} \frac{1}{kx} ∏_{p|k} (1 − \frac{1}{p})^{−1} \frac{x}{log^{2} x} ≪ \frac{log log x}{log^{2} x}.

¿From this it follows that the number of primes p with x < p ≤ 2x with

∑_{q≡±1modp} \frac{c(q)}{q} > \frac{c_{1}}{p log log p}

is ≪ x(log log x)^{2}/ log^{2} x. Using a dyadic decomposition again we get at most ≪ x(log log x)^{2}/ log^{2} x exceptional primes p, p ≤ x.

An application of Lemma 2.2 similar to the previous section shows that for all but O(x/ log^{B} x) primes p with p ≤ x we have

∑_{2^{A}p log^{A} p<q<p^{1+δ}} \frac{c(q)}{q} ≪ ∫_{2^{A}p log^{A} p}^{p^{1+δ}} \frac{dt}{pt log p} ≪ \frac{δ log p}{p log p}.

5. Concluding Remarks

It is clear from the previous sections that Kummer’s conjecture is related to the difference of the number of primes congruent to 1 and −1 modulo p.

¿From the analysis of the previous sections we see that Kummer’s conjecture for almost all primes is a consequence of the following condition (unproven)

∫_{p log^{A} p}^{p^{2}} \frac{E_{1}(t, p)}{t^{2}}dt = o(1/p),

where E_{1}(x, d) = π(x, d, 1) − π(x, d, −1).

Remark 5.1. L. Goldstein [6] proved the analogue of (1.4) for Q(ζ_{p^{r}}). As r → ∞ and for fixed odd p

log h_{p^{r}}^{−} ∼ \frac{1}{4} (1 − \frac{1}{p}) p^{r}r log p.

In this case the analogue of (2.1) is

h_{p^{r}}^{−} = \frac{p^{r+1/4+t(p,r)/4}}{2^{φ(p^{r})/2−1}π^{φ(p^{r})/2}} ∏_{χ(−1)=−1} L(1, χ) (5.1)

with t(p, r) = log |d_{Q(ζ_{p^{r}})}|/ log p = rp^{r} − (r + 1)p^{r−1}, and the product is over the odd characters of the Galois group of Q(ζ_{p^{r}}). It is interesting to investigate this product as r → ∞.

References

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Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada K7L 3N6

E-mail address: murty@mast.queensu.ca

Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada K7L 3N6

Current address: Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 2K6

E-mail address: petridis@math.mcgill.ca