[PPT] - Large and small G al sums Nombres premiers, d eterminisme et PowerPoint Presentation

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Large and small G´ al sums

Nombres premiers, d´ eterminisme et pseudoal´ ea CIRM, 4-8 novembre 2019 8/11/2019

G´ erald Tenenbaum Institut ´ Elie Cartan Universit´ e de Lorraine BP 70239 54506 Vandœuvre-l` es-Nancy Cedex France gerald.tenenbaum@univ-lorraine.fr

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– 1 –

1. G´

al sums

SLIDE 3

– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0),

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– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n.

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– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n. Koksma’s conjecture (1930’s): Γ1(N) := sup

|M|=N

S1(M)/N ≪ (log2 N)2.

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– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n. Koksma’s conjecture (1930’s): Γ1(N) := sup

|M|=N

S1(M)/N ≪ (log2 N)2. Key point : no bound on the size of m ∈ M, only on the size of |M|.

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– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n. Koksma’s conjecture (1930’s): Γ1(N) := sup

|M|=N

S1(M)/N ≪ (log2 N)2. Key point : no bound on the size of m ∈ M, only on the size of |M|. Erd˝

s (1947) : proposed a prize at the Amsterdam Math. Soc.:

SLIDE 8

– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n. Koksma’s conjecture (1930’s): Γ1(N) := sup

|M|=N

S1(M)/N ≪ (log2 N)2. Key point : no bound on the size of m ∈ M, only on the size of |M|. Erd˝

s (1947) : proposed a prize at the Amsterdam Math. Soc.:

Proved by G´ al (1949).

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– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n. Koksma’s conjecture (1930’s): Γ1(N) := sup

|M|=N

S1(M)/N ≪ (log2 N)2. Key point : no bound on the size of m ∈ M, only on the size of |M|. Erd˝

s (1947) : proposed a prize at the Amsterdam Math. Soc.:

Proved by G´ al (1949). Lewko & Radziwi l l (2017): Γ1(N) ∼ 6e2γ π2 (log2 N)2 (N → ∞).

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– 1 –

1. G´

al sums

M ⊂ N∗, |M| < ∞. G´ al sums: Sα(M) :=

m,n∈M

(m, n)α [m, n]α (α > 0), where (m, n) (resp. [m, n]) denotes the gcd (resp. the lcm) of m and n. Koksma’s conjecture (1930’s): Γ1(N) := sup

|M|=N

S1(M)/N ≪ (log2 N)2. Key point : no bound on the size of m ∈ M, only on the size of |M|. Erd˝

s (1947) : proposed a prize at the Amsterdam Math. Soc.:

Proved by G´ al (1949). Lewko & Radziwi l l (2017): Γ1(N) ∼ 6e2γ π2 (log2 N)2 (N → ∞). Applications: distribution modulo 1 of sequences {nkϑ}∞

k=1 for almost all ϑ:

1

m∈M

cmB(mx)

2

dx =

1 12

m,n∈M

(m, n)α [m, n]α cmcn (c ∈ C|M|).

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– 2 –

2. Bounding large G´

al sums

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– 2 –

2. Bounding large G´

al sums

Recent works: α = 1

2 — applications to zeta function and character sums.

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– 2 –

2. Bounding large G´

al sums

Recent works: α = 1

2 — applications to zeta function and character sums.

Resonance method.

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– 2 –

2. Bounding large G´

al sums

Recent works: α = 1

2 — applications to zeta function and character sums.

Resonance method. Improving Bondarenko and Seip (’15, ’17): Theorem 1 (La Bret` eche-T. 2018). Let L(N) := e √

(log Nlog3 N)/ log2 N.

Then Γ1/2(N) := max

|M|=N

S(M) |M| = L(N)2

√ 2+o(1).

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– 2 –

2. Bounding large G´

al sums

Recent works: α = 1

2 — applications to zeta function and character sums.

Resonance method. Improving Bondarenko and Seip (’15, ’17): Theorem 1 (La Bret` eche-T. 2018). Let L(N) := e √

(log Nlog3 N)/ log2 N.

Then Γ1/2(N) := max

|M|=N

S(M) |M| = L(N)2

√ 2+o(1).

The same estimate holds also for Q(M) := sup

c∈CN c2=1

m,n∈M

cmcn

(m, n)

[m, n]

.

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– 2 –

2. Bounding large G´

al sums

Recent works: α = 1

2 — applications to zeta function and character sums.

Resonance method. Improving Bondarenko and Seip (’15, ’17): Theorem 1 (La Bret` eche-T. 2018). Let L(N) := e √

(log Nlog3 N)/ log2 N.

Then Γ1/2(N) := max

|M|=N

S(M) |M| = L(N)2

√ 2+o(1).

The same estimate holds also for Q(M) := sup

c∈CN c2=1

m,n∈M

cmcn

(m, n)

[m, n]

.

BS consider subsums of G´ al type: S(M) :=

m,n∈M, n|m

n m 1/2 .

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– 2 –

2. Bounding large G´

al sums

Recent works: α = 1

2 — applications to zeta function and character sums.

Resonance method. Improving Bondarenko and Seip (’15, ’17): Theorem 1 (La Bret` eche-T. 2018). Let L(N) := e √

(log Nlog3 N)/ log2 N.

Then Γ1/2(N) := max

|M|=N

S(M) |M| = L(N)2

√ 2+o(1).

The same estimate holds also for Q(M) := sup

c∈CN c2=1

m,n∈M

cmcn

(m, n)

[m, n]

.

BS consider subsums of G´ al type: S(M) :=

m,n∈M, n|m

n m 1/2 . It can be shown (BS 2017, LB-T 2018) that max

|M|=N S(M) = L(N)o(1) while

the norm of the corresponding quadratic form is L(N)1+o(1).

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– 3 –

3. Applications

3·1. Localised maxima of |ζ( 1

2 + iτ)|

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– 3 –

3. Applications

3·1. Localised maxima of |ζ( 1

2 + iτ)|

Zβ(T) := max

T βτT

ζ( 1

2 + iτ)

(0 β < 1, T 1)

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– 3 –

3. Applications

3·1. Localised maxima of |ζ( 1

2 + iτ)|

Zβ(T) := max

T βτT

ζ( 1

2 + iτ)

(0 β < 1, T 1)

LB-T (2018): Zβ(T) L(T) √

2(1−β)+o(1).

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– 3 –

3. Applications

3·1. Localised maxima of |ζ( 1

2 + iτ)|

Zβ(T) := max

T βτT

ζ( 1

2 + iτ)

(0 β < 1, T 1)

LB-T (2018): Zβ(T) L(T) √

2(1−β)+o(1).

Improvement of Bondarenko and Seip’s exponent by a factor √ 2.

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– 3 –

3. Applications

3·1. Localised maxima of |ζ( 1

2 + iτ)|

Zβ(T) := max

T βτT

ζ( 1

2 + iτ)

(0 β < 1, T 1)

LB-T (2018): Zβ(T) L(T) √

2(1−β)+o(1).

Improvement of Bondarenko and Seip’s exponent by a factor √ 2.

3·2. Central values of L-functions

L(s, χ) :=

n1 χ(n)/ns

(χ = χ0, ℜe (s) > 0).

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– 3 –

3. Applications

3·1. Localised maxima of |ζ( 1

2 + iτ)|

Zβ(T) := max

T βτT

ζ( 1

2 + iτ)

(0 β < 1, T 1)

LB-T (2018): Zβ(T) L(T) √

2(1−β)+o(1).

Improvement of Bondarenko and Seip’s exponent by a factor √ 2.

3·2. Central values of L-functions

L(s, χ) :=

n1 χ(n)/ns

(χ = χ0, ℜe (s) > 0). LB-T (2018) : When q is prime and tends to ∞, max

χ mod q χ=χ0 χ(−1)=1

L( 1

2, χ)

L(q)1+o(1) = exp
1 + o(1)
log q log3 q

log2 q

.

Improves Soundararajan (2008), Hough (2016), by an extra factor ≍

log3 q.

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– 4 –

3·3. Character sums

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– 4 –

3·3. Character sums

∆(x, q) := max χ=χ0

χ mod q

nx χ(n)
,

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– 4 –

3·3. Character sums

∆(x, q) := max χ=χ0

χ mod q

nx χ(n)
,

LB-T (2018): When e(log q)1/2+ε x q/e(1+ε)ω(q), we have ∆(x, q) ≫ √xL(q/x)

√ 2+o(1)

(q → ∞).

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– 4 –

3·3. Character sums

∆(x, q) := max χ=χ0

χ mod q

nx χ(n)
,

LB-T (2018): When e(log q)1/2+ε x q/e(1+ε)ω(q), we have ∆(x, q) ≫ √xL(q/x)

√ 2+o(1)

(q → ∞). In its range, improves Hough’s estimate (2013) by an extra factor

log3(q/x).

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– 4 –

3·3. Character sums

∆(x, q) := max χ=χ0

χ mod q

nx χ(n)
,

LB-T (2018): When e(log q)1/2+ε x q/e(1+ε)ω(q), we have ∆(x, q) ≫ √xL(q/x)

√ 2+o(1)

(q → ∞). In its range, improves Hough’s estimate (2013) by an extra factor

log3(q/x).

Valid not only for q prime.

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– 5 –

4. Small G´

al sums

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– 5 –

4. Small G´

al sums

T(c; N) :=

m,nN

(m, n) √mn cmcn, TN := N inf

c∈(R+)N c1=1

T(c; N), V(c; N) :=

m,nN

(m, n) m + ncmcn, VN := N inf

c∈(R+)N c1=1

V(c; N),

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– 5 –

4. Small G´

al sums

T(c; N) :=

m,nN

(m, n) √mn cmcn, TN := N inf

c∈(R+)N c1=1

T(c; N), V(c; N) :=

m,nN

(m, n) m + ncmcn, VN := N inf

c∈(R+)N c1=1

V(c; N), Trivial bounds : VN 1

2TN ≪ (log N).

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– 5 –

4. Small G´

al sums

T(c; N) :=

m,nN

(m, n) √mn cmcn, TN := N inf

c∈(R+)N c1=1

T(c; N), V(c; N) :=

m,nN

(m, n) m + ncmcn, VN := N inf

c∈(R+)N c1=1

V(c; N), Trivial bounds : VN 1

2TN ≪ (log N).

Theorem 2 (La Bret` eche-Munsch-T, 2019). Let η := 0.16656 . . . < 1/6. Then (log N)η ≪ VN 1

2TN ≪ (log N)η(log2 N)3.

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– 5 –

4. Small G´

al sums

T(c; N) :=

m,nN

(m, n) √mn cmcn, TN := N inf

c∈(R+)N c1=1

T(c; N), V(c; N) :=

m,nN

(m, n) m + ncmcn, VN := N inf

c∈(R+)N c1=1

V(c; N), Trivial bounds : VN 1

2TN ≪ (log N).

Theorem 2 (La Bret` eche-Munsch-T, 2019). Let η := 0.16656 . . . < 1/6. Then (log N)η ≪ VN 1

2TN ≪ (log N)η(log2 N)3.

η is the solution of some transcendal equation.

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– 6 – Application: an improvement of Burgess’ bound

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

P´
lya–Vinogradov (1918): S ≪ √p log p, non trivial for N > p1/2+ε.

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

P´
lya–Vinogradov (1918): S ≪ √p log p, non trivial for N > p1/2+ε.
Burgess (1962): (∗) S ≪ N 1−1/rp(r+1)/4r2(log p)b (r 1) with b = 1.

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

P´
lya–Vinogradov (1918): S ≪ √p log p, non trivial for N > p1/2+ε.
Burgess (1962): (∗) S ≪ N 1−1/rp(r+1)/4r2(log p)b (r 1) with b = 1.

Non trivial for N > p1/4+ε.

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

P´
lya–Vinogradov (1918): S ≪ √p log p, non trivial for N > p1/2+ε.
Burgess (1962): (∗) S ≪ N 1−1/rp(r+1)/4r2(log p)b (r 1) with b = 1.

Non trivial for N > p1/4+ε.

Kerr, Shparlinski & Yau (2017): b = 1/4r + o(1).

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

P´
lya–Vinogradov (1918): S ≪ √p log p, non trivial for N > p1/2+ε.
Burgess (1962): (∗) S ≪ N 1−1/rp(r+1)/4r2(log p)b (r 1) with b = 1.

Non trivial for N > p1/4+ε.

Kerr, Shparlinski & Yau (2017): b = 1/4r + o(1).

Theorem 3 (La Bret` eche, Munsch, T. 2019). For r 1, p 1

2N, we have S ≪ N 1−1/rp(r+1)/4r2 max 1xp T1/2r x

.

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– 6 – Application: an improvement of Burgess’ bound S = S(M, N; χ) :=

M<nM+N

χ(n) (χ Dirichlet character mod p).

P´
lya–Vinogradov (1918): S ≪ √p log p, non trivial for N > p1/2+ε.
Burgess (1962): (∗) S ≪ N 1−1/rp(r+1)/4r2(log p)b (r 1) with b = 1.

Non trivial for N > p1/4+ε.

Kerr, Shparlinski & Yau (2017): b = 1/4r + o(1).

Theorem 3 (La Bret` eche, Munsch, T. 2019). For r 1, p 1

2N, we have S ≪ N 1−1/rp(r+1)/4r2 max 1xp T1/2r x

. Hence (∗) holds for b = η/2r + o(1) < 1/12r.

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– 7 –

5. Multiplicative energy

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– 7 –

5. Multiplicative energy

Gowers (1998), Tao-Vu (2006), Tao (2008): E(A, B) :=

n1

ab=n

1 2 .

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– 7 –

5. Multiplicative energy

Gowers (1998), Tao-Vu (2006), Tao (2008): E(A, B) :=

n1

ab=n

1 2 . Easy : E([1, N], B) ≍ N

b,b′∈B

(b, b′) b + b′ .

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– 7 –

5. Multiplicative energy

Gowers (1998), Tao-Vu (2006), Tao (2008): E(A, B) :=

n1

ab=n

1 2 . Easy : E([1, N], B) ≍ N

b,b′∈B

(b, b′) b + b′ . Weighted version: E(c; N) :=

1nN 2

dt=n d,tN

cdct 2 , EN := inf

c∈(R+)N c1=1

N 2E(c; N).

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– 7 –

5. Multiplicative energy

Gowers (1998), Tao-Vu (2006), Tao (2008): E(A, B) :=

n1

ab=n

1 2 . Easy : E([1, N], B) ≍ N

b,b′∈B

(b, b′) b + b′ . Weighted version: E(c; N) :=

1nN 2

dt=n d,tN

cdct 2 , EN := inf

c∈(R+)N c1=1

N 2E(c; N). Let δ := 1 − (1 + log2 2)/ log 2 ≈ 0.08607 < 5

58, exponent appearing in Erd˝

s’

multiplication table problem, Erd˝

s ’60, T ’80-’84, Hall-T ’88, Ford ’06:

H(N) :=

n N 2 : ∃a, b N, n = ab
≍

N 2 (log N)δ(log2 N)3/2 · Theorem 4 (La Bret` eche, Munsch & T, 2019). (log N)δ(log2 N)3/2 ≪ EN ≪ (log N)δ(log2 N)6 (N 3).

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– 8 – Application 1: non vanishing of theta functions

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– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

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– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

Consider ϑ(x; χ) :=

n1

χ(n)e−πn2x/p (χ ∈ X+

p := {χ (mod p) : χ(−1) = 1}).

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– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

Consider ϑ(x; χ) :=

n1

χ(n)e−πn2x/p (χ ∈ X+

p := {χ (mod p) : χ(−1) = 1}).

M0(p) = {χ ∈ X+

p : ϑ(1; χ) = 0}.

SLIDE 51

– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

Consider ϑ(x; χ) :=

n1

χ(n)e−πn2x/p (χ ∈ X+

p := {χ (mod p) : χ(−1) = 1}).

M0(p) = {χ ∈ X+

p : ϑ(1; χ) = 0}.

Louboutin conjectured M0(p) = 1

2(p − 1).

SLIDE 52

– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

Consider ϑ(x; χ) :=

n1

χ(n)e−πn2x/p (χ ∈ X+

p := {χ (mod p) : χ(−1) = 1}).

M0(p) = {χ ∈ X+

p : ϑ(1; χ) = 0}.

Louboutin conjectured M0(p) = 1

2(p − 1).

Checked for 3 p 106 by Molin ’18.

SLIDE 53

– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

Consider ϑ(x; χ) :=

n1

χ(n)e−πn2x/p (χ ∈ X+

p := {χ (mod p) : χ(−1) = 1}).

M0(p) = {χ ∈ X+

p : ϑ(1; χ) = 0}.

Louboutin conjectured M0(p) = 1

2(p − 1).

Checked for 3 p 106 by Molin ’18. Louboutin & Munsch ’13 : M0(p) ≫ p/ log p.

SLIDE 54

– 8 – Application 1: non vanishing of theta functions Balasubramanian & Murty ’92: L( 1

2, χ) = 0 for a positive proportion of characters.

Consider ϑ(x; χ) :=

n1

χ(n)e−πn2x/p (χ ∈ X+

p := {χ (mod p) : χ(−1) = 1}).

M0(p) = {χ ∈ X+

p : ϑ(1; χ) = 0}.

Louboutin conjectured M0(p) = 1

2(p − 1).

Checked for 3 p 106 by Molin ’18. Louboutin & Munsch ’13 : M0(p) ≫ p/ log p. Theorem 5 (La Bret` eche, Munsch & T, 2019). Put q :=

p/3
. Then

M0(p) ≫ p Eq ≫ p (log p)δ(log2 p)6 .

SLIDE 55

– 9 – Application 2: Lower bounds for low moments of character sums Recently, Harper ’17 announced 1 p − 2

χ=χ0
nN

χ(n)

≪

√ N min (log2 ν)1/4 with ν := min {N, p/N}. More than square root cancellation!

SLIDE 56

– 9 – Application 2: Lower bounds for low moments of character sums Recently, Harper ’17 announced 1 p − 2

χ=χ0
nN

χ(n)

≪

√ N min (log2 ν)1/4 with ν := min {N, p/N}. More than square root cancellation! Theorem 6 (BMT 2019). Let r ∈]0, 4/3[. For large p and all N ∈ [1, 1

2p[,

we have 1 p − 2

χ=χ0
S(N; χ)
r

≫ N r/2 E1−r/2

ν

.

SLIDE 57

– 10 – Application 2: Lower bounds for low moments of character sums Recently, Harper ’17 announced 1 p − 2

χ=χ0
nN

χ(n)

≪

√ N min (log2 ν)1/4 with ν := min {N, p/N}. More than square root cancellation! Theorem 6 (BMT 2019). Let r ∈]0, 4/3[. For large p and all N ∈ [1, 1

2p[,

we have 1 p − 2

χ=χ0
S(N; χ)
r

≫ N r/2 E1−r/2

ν

. In particular, 1 p − 2

χ=χ0
S(N; χ)
≫
N

Eν ≫ √ N (log ν)δ/2(log2 ν)3 . Note : 1

2δ ≈ 0, 04303.

SLIDE 58

– 11 – Same method yields: 1 T T

nN

nit

r

dt ≫ N r/2 E1−r/2

ν

(T 1, 1 N √ T)

SLIDE 59

– 11 – Same method yields: 1 T T

nN

nit

r

dt ≫ N r/2 E1−r/2

ν

(T 1, 1 N √ T) In particular, lim

T →∞

1 T T

nN

nit

dt ≫
N

EN ≫ √ N (log N)δ/2(log2 N)3 .

SLIDE 60

– 11 – Same method yields: 1 T T

nN

nit

r

dt ≫ N r/2 E1−r/2

ν

(T 1, 1 N √ T) In particular, lim

T →∞

1 T T

nN

nit

dt ≫
N

EN ≫ √ N (log N)δ/2(log2 N)3 . Improves on Bondarenko & Seip (2016), who obtained an exponent ≈ 0.05616 by a different approach, which does not extend to character sums.

SLIDE 61

– 12 –

6. Proofs for E(c; N)

The lower bound follows from the Cauchy-Schwarz inequality:

SLIDE 62

– 12 –

6. Proofs for E(c; N)

The lower bound follows from the Cauchy-Schwarz inequality: r(n) :=

dt=n, d,tN

cdct,

SLIDE 63

– 12 –

6. Proofs for E(c; N)

The lower bound follows from the Cauchy-Schwarz inequality: r(n) :=

dt=n, d,tN

cdct, c4

1 = nN 2

r(n) 2 H(N)

nN 2

r(n)2 = H(N)E(c; N) where H(N) :=

n N 2 : ∃a, b N, n = ab
.

SLIDE 64

– 12 –

6. Proofs for E(c; N)

The lower bound follows from the Cauchy-Schwarz inequality: r(n) :=

dt=n, d,tN

cdct, c4

1 = nN 2

r(n) 2 H(N)

nN 2

r(n)2 = H(N)E(c; N) where H(N) :=

n N 2 : ∃a, b N, n = ab
.

By Ford ’06 : H(N) ≪ N 2/

(log N)δ(log2 N)3/2

.

SLIDE 65

– 13 – To establish the upper bound, select n → cn as the indicator function of the set of those integers n ∈ ] 3

4N, N] satisfying

(∗) Ω(n) = log2 N log 4

,

Ω(n; t) log2(3t) log 4 + C (1 t N).

SLIDE 66

– 13 – To establish the upper bound, select n → cn as the indicator function of the set of those integers n ∈ ] 3

4N, N] satisfying

(∗) Ω(n) = log2 N log 4

,

Ω(n; t) log2(3t) log 4 + C (1 t N). For large enough C, we may adapt an argument of Ford (2007) to show

m1

r(m) = c2

1 ≫

N (log N)δ(log2 N)3 ·

SLIDE 67

– 13 – To establish the upper bound, select n → cn as the indicator function of the set of those integers n ∈ ] 3

4N, N] satisfying

(∗) Ω(n) = log2 N log 4

,

Ω(n; t) log2(3t) log 4 + C (1 t N). For large enough C, we may adapt an argument of Ford (2007) to show

m1

r(m) = c2

1 ≫

N (log N)δ(log2 N)3 · Thus N 2 c4

1

E(c; N) = N 2 c4

1

nN 2

r(n)2 ≪ N 4 c4

1(log N)δ ·

SLIDE 68

– 13 – To establish the upper bound, select n → cn as the indicator function of the set of those integers n ∈ ] 3

4N, N] satisfying

(∗) Ω(n) = log2 N log 4

,

Ω(n; t) log2(3t) log 4 + C (1 t N). For large enough C, we may adapt an argument of Ford (2007) to show

m1

r(m) = c2

1 ≫

N (log N)δ(log2 N)3 · Thus N 2 c4

1

E(c; N) = N 2 c4

1

nN 2

r(n)2 ≪ N 4 c4

1(log N)δ ·

The last upper bound follows from an inequality in Divisors (Hall-T. ’88) on noticing that any n counted in the sum satisfies (∗) with log 2 in place of log 4.

SLIDE 69

– 14 –

7. Proofs for T(c;N).

SLIDE 70

– 14 –

7. Proofs for T(c;N).

We only consider the lower bound. Since (m, n) =

d|m,n ϕ(d), we get

T(c; N) =

m,nN

(m, n) √mn cmcn =

dN

ϕ(d) d x2

d,

with c := {cn}N

n=1, c1 = 1, and xd :=

mN/d

cmd √m·

SLIDE 71

– 14 –

7. Proofs for T(c;N).

We only consider the lower bound. Since (m, n) =

d|m,n ϕ(d), we get

T(c; N) =

m,nN

(m, n) √mn cmcn =

dN

ϕ(d) d x2

d,

with c := {cn}N

n=1, c1 = 1, and xd :=

mN/d

cmd √m· Let β, y ∈]0, 1]. We have S :=

dN

yΩ(d) xd √ d =

nN

cn √n

d|n

yΩ(d)

nN

cn(1 + y)ω(n) √n .

SLIDE 72

– 14 –

7. Proofs for T(c;N).

We only consider the lower bound. Since (m, n) =

d|m,n ϕ(d), we get

T(c; N) =

m,nN

(m, n) √mn cmcn =

dN

ϕ(d) d x2

d,

with c := {cn}N

n=1, c1 = 1, and xd :=

mN/d

cmd √m· Let β, y ∈]0, 1]. We have S :=

dN

yΩ(d) xd √ d =

nN

cn √n

d|n

yΩ(d)

nN

cn(1 + y)ω(n) √n . If ω(n) β log2 N (n ∈ A) with

n∈A

cn 1

2, then S ≫ (log N)β log(1+y)/

√ N, ... (log N)2β log(1+y) N ≪ S2 ≪

dN

ϕ(d) d x2

d

dN

y2Ω(d) ϕ(d) ≪ T(c; N)(log N)y2,

SLIDE 73

– 14 –

7. Proofs for T(c;N).

We only consider the lower bound. Since (m, n) =

d|m,n ϕ(d), we get

T(c; N) =

m,nN

(m, n) √mn cmcn =

dN

ϕ(d) d x2

d,

with c := {cn}N

n=1, c1 = 1, and xd :=

mN/d

cmd √m· Let β, y ∈]0, 1]. We have S :=

dN

yΩ(d) xd √ d =

nN

cn √n

d|n

yΩ(d)

nN

cn(1 + y)ω(n) √n . If ω(n) β log2 N (n ∈ A) with

n∈A

cn 1

2, then S ≫ (log N)β log(1+y)/

√ N, ... (log N)2β log(1+y) N ≪ S2 ≪

dN

ϕ(d) d x2

d

dN

y2Ω(d) ϕ(d) ≪ T(c; N)(log N)y2, and so TN NT(c; N) ≫ (log N)η, with f(β) := miny{2β log(1 + y) − y2}.

SLIDE 74

– 15 – If, on the contrary,

SLIDE 75

– 15 – If, on the contrary,

n∈B

cn 1

2, with B := {n N : ω(n) β log2 N},

SLIDE 76

– 15 – If, on the contrary,

n∈B

cn 1

2, with B := {n N : ω(n) β log2 N},

then

d∈B

xd =

nN

cn √n

d|n, d∈B

√ d

n∈B

cn 1

2.

SLIDE 77

– 15 – If, on the contrary,

n∈B

cn 1

2, with B := {n N : ω(n) β log2 N},

then

d∈B

xd =

nN

cn √n

d|n, d∈B

√ d

n∈B

cn 1

2.

... 1 ≪

d∈B

d ϕ(d)

dN

ϕ(d) d x2

d ≪

T(c; N)N (log N)Q(β) log2 N

TN

(log N)Q(β) , with Q(β) := β log β − β + 1.

SLIDE 78

– 15 – If, on the contrary,

n∈B

cn 1

2, with B := {n N : ω(n) β log2 N},

then

d∈B

xd =

nN

cn √n

d|n, d∈B

√ d

n∈B

cn 1

2.

... 1 ≪

d∈B

d ϕ(d)

dN

ϕ(d) d x2

d ≪

T(c; N)N (log N)Q(β) log2 N

TN

(log N)Q(β) , with Q(β) := β log β − β + 1. Select β such that f(β) = Q(β) = η, we get TN ≫ (log N)η.

SLIDE 79

– 16 –

8. Proof for mean values of character sums

SLIDE 80

– 16 –

8. Proof for mean values of character sums

Given c ∈ (R+)N, define M(N; χ) =

mN

cmχ(m), and

SLIDE 81

– 16 –

8. Proof for mean values of character sums

Given c ∈ (R+)N, define M(N; χ) =

mN

cmχ(m), and Sk(N) := 1 p − 1

χ=χ0

|S(N; χ)|k (k > 0), M4(N) := 1 p − 1

χ=χ0

|M(N; χ)|4. H¨

lder’s inequality yields

c1 ≪ 1 p − 1

χ=χ0

S(N; χ)M(N; χ)

S1(N)1/2S2(N)1/4M4(N)1/4.

SLIDE 82

– 16 –

8. Proof for mean values of character sums

Given c ∈ (R+)N, define M(N; χ) =

mN

cmχ(m), and Sk(N) := 1 p − 1

χ=χ0

|S(N; χ)|k (k > 0), M4(N) := 1 p − 1

χ=χ0

|M(N; χ)|4. H¨

lder’s inequality yields

c1 ≪ 1 p − 1

χ=χ0

S(N; χ)M(N; χ)

S1(N)1/2S2(N)1/4M4(N)1/4.

Now M4(N) ≪ E(c; N) and S2(N) ≪ N by orthogonality.

SLIDE 83

– 16 –

8. Proof for mean values of character sums

Given c ∈ (R+)N, define M(N; χ) =

mN

cmχ(m), and Sk(N) := 1 p − 1

χ=χ0

|S(N; χ)|k (k > 0), M4(N) := 1 p − 1

χ=χ0

|M(N; χ)|4. H¨

lder’s inequality yields

c1 ≪ 1 p − 1

χ=χ0

S(N; χ)M(N; χ)

S1(N)1/2S2(N)1/4M4(N)1/4.

Now M4(N) ≪ E(c; N) and S2(N) ≪ N by orthogonality. By choosing c optimally, we deduce S1(N) ≫ c2

1

E(c; N)1/2N 1/2 ≫ N 1/2 E1/2

N

·

SLIDE 84