On Gelfond’s conjecture about the sum of digits of prime numbers

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Joël RIVAT

On Gelfond’s conjecture about the sum of digits of prime numbers Tome 21, no2 (2009), p. 415-422.

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de Bordeaux 21 (2009), 415-422

On Gelfond’s conjecture about

the sum of digits of prime numbers

par Joël RIVAT

Résumé. Dans cet article nous exposons les étapes importantes de la preuve de la conjecture de Gelfond [6] (1968) dans un travail récent en collaboration avec Christian Mauduit [11] concernant la somme des chiffres des nombres premiers, dans l’esprit de l’ex-posé donné à Edimbourg dans le cadre des Journées Arithmétiques 2007.

Abstract. The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

1. The sum of digits function

Let q ∈ N with q > 2. All n ∈ N can be written uniquely in basis q: n =X

k>0

nkqk where nk∈ {0, . . . , q − 1}

and the sum of digits function is defined by: s(n) = X

k>0

nk.

The sum of digits function has many aspects that have been studied, for instance ergodicity, finite automata, dynamical systems, number theory.

Mahler introduced this function in the context of harmonic analysis:

Theorem A (Mahler [10], 1927). For q = 2, the sequence

1 N X n<N (−1)s(n)(−1)s(n+k) ! N >1

converges for all k ∈ N and its limit is different from zero for infinitely many k’s.

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416 _{Joël Rivat}

Theorem B (Gelfond [6], 1968). Let m > 2, (m, q − 1) = 1. Then there

exists λ < 1 such that for all d ∈ N∗, a, r ∈ Z,

X n<N n≡r mod d s(n)≡a mod m 1 = N md+ O(N λ_).

In the same paper Gelfond pose the following problem:

Problem A (Gelfond [6], 1968).

Il serait aussi intéressant de trouver le nombre des nombres premiers p 6 x tels que s(p) ≡ a mod m.

(the letter p always denotes a prime number).

2. Prime numbers

The prime numbers constitute a fascinating sequence which poses many difficult questions. Let us mention some open problems:

• are there infinitely many primes of the form p + 2 (prime twins)? • are there infinitely many primes of the form n2_{+ 1?}

• are there infinitely many primes of the form 2n _{− 1 (Mersenne}

primes)?

• are there infinitely many primes of the form 22n

+1 (Fermat primes)? and some answers:

• primes of the form an + b (Dirichlet theorem),

• primes such that αp belongs to some prescribed interval I ⊂ [0, 1], for α ∈ R \ Q (Vinogradov theorem [15]),

• primes of the form bnc_{c where 1 < c < c}

0 ≈ 1.1 (Piatetski-Shapiro

theorem [12]),

• primes of the form a2_{+ b}4 _{(Friedlander-Iwaniec theorem [4, 5]),}

• primes of the form a3_{+ 2b}3 _{(Heath-Brown theorem [9]),}

• primes such that a_p(E)/(2√p) belongs to some prescribed interval I ⊂ [0, 1] (Sato-Tate conjecture for a very large class of elliptic curves E over Q, recently proved by several authors).

3. Historical background

Until recently, very little was known concerning the digits of prime num-bers. We can mention a result of Sierpiński [13] (1959), recently generalized by Wolke [16] (2005) and then by Harman [7] (2006), on prime numbers with some prescribed digits.

Concerning Gelfond’s question, no progress was made in its original form. Let us mention the two following variants:

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Theorem C (Fouvry–Mauduit [2, 3], 1996). For m > 2 such that (m, q − 1) = 1, there exists C(q, m) > 0 such that for all a ∈ Z and x > 0,

X n6x n=p or n=p1p2 s(n)≡a mod m 1 > C(q, m) log log x X n6x n=p or n=p1p2 1.

Theorem D (Dartyge–Tenenbaum [1], 2005). For m_{> 2 with (m, q −1) =}

1 and r > 2, there exists C(q, m, r) > 0 such that for all a ∈ Z and x > 0,

X

n6x n=p1...pr

s(n)≡a mod m

1 > C(q, m, r) log log x log log log x

X

n6x n=p1...pr

1.

4. Results

Theorem 1 (Mauduit-Rivat). For α ∈ R such that (q − 1)α ∈ R \ Z, there exists C(q, α) > 0 and σq(α) > 0, X p6x e(α s(p)) 6 C(q, α) x1−σq(α)

where e(t) = exp(2iπt).

Corollary 1. The sequence (α s(p_n))_n>1 is equidistributed modulo 1 if and only if α ∈ R \ Q (here (pn)_n>1 denotes the sequence of prime numbers). Corollary 2. For m_{> 2 such that (m, q − 1) = 1 and a ∈ Z,}

X p6x s(p)≡a mod m 1 ∼ 1 m X p6x 1 (x → +∞).

5. Sum over prime numbers

We want to estimate a sum of the form

X

p6x

g(p)

where the function g detects the property under consideration. A classical process (Vinogradov [15], Vaughan [14], Heath-Brown [8]) remains (using some more technical details), for some 0 < β1 < 1/3 and 1/2 < β2 < 1, to

estimate uniformly the sums SI := X m∼M X n∼N g(mn) for M 6 xβ1 _{(type I)}

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418 _{Joël Rivat}

where M N = x (which implies that the “easy” sum over n is long) and for all complex numbers am, bn with |am| 6 1, |bn| 6 1 the sums

SII :=

X

m∼M

X

n∼N

ambng(mn) for xβ1 < M 6 xβ2 (type II),

(which implies that both sums have a significant length).

6. Sums of type I

For the sums of type I we might expect that the knowledge of the function g permits to get a satisfactory estimate of the sum

X

n∼N

g(mn).

Indeed in our case where g(n) = e(α s(n)) we were able to adapt success-fully arguments from Fouvry and Mauduit [2, 3] (1996).

7. Sums of type II 7.1. Smoothing the sums.

By Cauchy-Schwarz inequality: |S_II|2_{6 M} X m∼M X n∼N bn e(α s(mn)) 2 .

Here, expanding the square and exchanging the summations, we would get a smooth sum over m, but also two free variables n1 and n2. However, we

can get a useful control by using van der Corput’s inequality:

Lemma 1. Let z1, . . . , zL∈ C. For all R ∈ N∗ we have

X 16`6L z` 2 6 L + R − 1 R X |r|<R 1 −|r| R X 16`6L 16`+r6L z`+rz`.

The interest of this inequality is that now we have n1= n + r and n2= n

so that the size of n1− n2= r is under control.

Now in fact we can take M = qµ, N = qν and R = qρ where µ, ν and ρ are integers such that ρ/(µ + ν) is “very small”. It remains for 1 6 |r| < qρ to prove the estimate

X qν−1_<n6qν bn+rbn X qµ−1_<m6qµ e(α s(m(n + r)) − α s(mn)) = O(qµ+ν−ρ).

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7.2. Truncated sum of digits function.

We want to take advantage of the fact that in the difference s(m(n + r)) − s(mn), the product mr is much smaller that mn. In the example: mn = µ+ν z }| { 35116790780999806546523475473462336857643565, mr = 396576345354568797095646467570 | {z } µ+ρ ,

we see that in the sum mn + mr the digits after index µ + ρ may change only by carry propagation.

Proving that the number of pairs (m, n) for which the carry propagation exceeds

λ := µ + 2ρ

is bounded by O(qµ+ν−ρ), we can ignore them and replace s(m(n + r)) − s(mn) by sλ(m(n + r)) − sλ(mn) where sλ is the truncated sum of digits

function

sλ(n) :=

X

k<λ

nk,

which is periodic of period qλ.

7.3. Fourier analysis.

The periodicity of sλ enables us to write

X qµ−1_<m6qµ e(α sλ(m(n + r)) − α sλ(mn)) = X 06u<qλ X 06v<qλ e(α sλ(u) − α sλ(v)) X qµ−1<m6qµ m(n+r)≡u mod qλ mn≡v mod qλ 1.

The orthogonality formula 1 qλ X 06h<qλ e _h` qλ = ( 1 if ` ≡ 0 mod qλ, 0 if ` 6≡ 0 mod qλ_,

leads us to introduce the discrete Fourier transform of u 7→ e(α sλ(u)):

Fλ(h) = q−λ X 06u<qλ e α sλ(u) − hu qλ ,

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420 _{Joël Rivat}

and summing over n and taking absolute values we must show that

X 06h<qλ X 06k<qλ Fλ(h) Fλ(−k) X qν−1_<n6qν X qµ−1_<m6qµ e _{hm(n + r) + kmn} qλ = O(qµ+ν−ρ). Here we observe that the summations over m (geometric sum !) and n can be handled by classical arguments from analytic number theory, while we hope that the digital structure hidden in Fλ will produce a huge saving. 7.4. Heuristic end of the proof.

On average, for fixed (h, k), the geometric sum over m is small so that the sum over n should be O(qν+ε). Hence after many technical steps to handle the exceptions, we will need to get the crucial upper bound

X

06h<qλ

|F_λ(h)| = Oqηλ with η < 1/2,

which means that we need an upper bound sharper than the square root of the trivial estimate.

Indeed suppose this has been done, then we get

X h X k X n X m · · · = O(q2ηλ+ν+ε_),

and since λ = µ + 2ρ, we have

2ηλ + ν + ε 6 µ + ν − ρ for µ, ν large enough.

8. The discrete Fourier transform

Let us take q = 2 to simplify all formulas. It follows from the definition of Fλ that

F0(h) = 1,

and that for λ > 1,

Fλ(h) = 1 2λ X 06u<2λ e α s(u) −uh 2λ so that Fλ(h) = 1 2λ X 06u<2λ−1 e α s(2u) −2uh 2λ + e α s(2u+1) − (2u+1)h 2λ . We have

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hence Fλ(h) = 1 2 1 + e α − h 2λ Fλ−1(h) thus |Fλ(h)| = λ Y i=1 cos π α − h 2i .

We recall that we want to prove that

X

06h<2λ

|F_λ(h)| = O(2ηλ) with η < 1₂.

9. Norm 1 of the discrete Fourier transform

Let the transfer operator Φ0(x) = 1 Φi(x) = cos π(α −x₂) Φ_i−1(x₂) + sin π(α − x₂) Φ_i−1(x+1₂ ). We write X 06h<2λ |Fλ(h)| = X 06h<2λ−1 |Fλ(h)| + Fλ(h + 2 λ−1₎ = X 06h<2λ−1 |Fλ−1(h)| cos π(α − h 2λ) + sin π(α − h 2λ) = X 06h<2λ−1 |Fλ−1(h)| Φ1 _h 2λ−1 . We can repeat this process and get

X 06h<2λ |F_λ(h)| = X 06h<2λ−2 |F_λ−2(h)| Φ2 _h 2λ−2 . Now Φ2₁(x) = cos π(α −x 2) + sin π(α − x 2) 2 = 1 + sin 2π(α −x 2) , and Φ2₂(x) 6 Φ21(x2) + Φ 2 1(x+12 ) = 2 +sin 2π(α − x₄) + sin 2π(α − x+1 4 ) = 2 + sin 2π(α − x₄) + cos 2π(α −x₄) , 6 2 +√2. Hence X 06h<2λ |Fλ(h)| 6 (2 + √ 2)1/2 X 06h<2λ−2 |Fλ−2(h)| ,

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422 _{Joël Rivat} and finally X 06h<2λ |Fλ(h)| = O (2 + √ 2)λ/4. Since (2 +√2)1/4< 41/4=√2, we indeed have X 06h<2λ |Fλ(h)| = O 2ηλ for some η < 1/2. References

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