the smoothed pólya vinogradov...
TRANSCRIPT
![Page 1: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/1.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
The Smoothed Pólya–Vinogradov Inequality
Enrique Treviño
Swarthmore College
Integers ConferenceOctober 28, 2011
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 2: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/2.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Pólya–Vinogradov
Let χ be a Dirichlet character to the modulus q > 1. Let
S(χ) = maxM,N
∣∣∣∣∣M+N∑
n=M+1
χ(n)
∣∣∣∣∣The Pólya–Vinogradov inequality (1918) states that there existsan absolute universal constant c such that for any Dirichletcharacter S(χ) ≤ c
√q log q.
Under GRH, Montgomery and Vaughan showed thatS(χ)� √q log log q.
Paley showed in 1932 that there are infinitely many quadraticcharacters such that S(χ)� √q log log q.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 3: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/3.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Further results regarding Pólya–Vinogradov
Granville and Soundararajan showed that one can save a smallpower of log q in the Pólya–Vinogradov inequality. Goldmakherimproved it to
Theorem (Goldmakher, 2007)For each fixed odd number g > 1, for χ (mod q) of order g,
S(χ)�g√
q(log q)∆g +o(1), ∆g =gπ
sinπ
g, q →∞.
Moreover, under GRHS(χ)�g
√q(log log q)∆g +o(1).
Furthermore, there exists an infinite family of characters χ (mod q) of order gsatisfying
S(χ)�ε,g√
q(log log q)∆g−ε.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 4: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/4.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Explicit Pólya–Vinogradov
Theorem (Hildebrand, 1988)For χ a primitive character to the modulus q > 1, we have
|S(χ)| ≤
(
23π2 + o(1)
)√
q log q , χ even,(1
3π+ o(1)
)√
q log q , χ odd .
Theorem (Pomerance, 2009)For χ a primitive character to the modulus q > 1, we have
|S(χ)| ≤
2π2
√q log q +
4π2
√q log log q +
32√
q , χ even,
12π√
q log q +1π
√q log log q +
√q , χ odd .
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 5: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/5.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Smoothed Pólya–Vinogradov
Let M,N be real numbers with 0 < N ≤ q, then define S∗(χ) asfollows:
S∗(χ) = maxM,N
∣∣∣∣∣∣∑
M≤n≤M+2N
χ(n)
(1−
∣∣∣∣n −MN
− 1∣∣∣∣)∣∣∣∣∣∣ .
Theorem (Levin, Pomerance, Soundararajan, 2009)Let χ be a primitive character to the modulus q > 1, and letM,N be real numbers with 0 < N ≤ q, then
S∗(χ) ≤√
q − N√
q.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 6: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/6.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Some Applications of the SmoothedPólya–Vinogradov
To prove a conjecture of Brizolis (Levin, Pomerance,Soundararajan) that for every prime p > 3 there is aprimitive root g and an integer x ∈ [1,p − 1] withlogg x = x , that is, gx ≡ x (mod p).Pólya–Vinogradov was used to prove a conjecture of Mollin(Granville, Mollin and Williams) that the least inert prime pin a real quadratic field of discriminant D is ≤
√D/2. We
used the Smoothed Pólya–Vinogradov to improve this top ≤ D0.45.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 7: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/7.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Proof of the Smoothed PV
Let H(t) = max{0,1− |t |}.
S∗(χ) =1
τ(χ)
q∑j=1
χ(j)∑n∈Z
e(nj/q)H(
n −MN
− 1).
The Fourier transform of H, H, is nonnegative andH(0) = 1.Using Poisson summation, the fact that χ(n) and e(n) haveabsolute value 1 and that |τ(χ)| =
√q for primitive
characters χ yields
|S∗(χ)| ≤ N√
q
∑k∈Z
(k ,q)=1
H(
kNq
).
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 8: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/8.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Proof of the Smoothed PV
Let H(t) = max{0,1− |t |}.
S∗(χ) =1
τ(χ)
q∑j=1
χ(j)∑n∈Z
e(nj/q)H(
n −MN
− 1).
The Fourier transform of H, H, is nonnegative andH(0) = 1.Using Poisson summation, the fact that χ(n) and e(n) haveabsolute value 1 and that |τ(χ)| =
√q for primitive
characters χ yields
|S∗(χ)| ≤ N√
q
∑k∈Z
(k ,q)=1
H(
kNq
).
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 9: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/9.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Proof of the Smoothed PV
Let H(t) = max{0,1− |t |}.
S∗(χ) =1
τ(χ)
q∑j=1
χ(j)∑n∈Z
e(nj/q)H(
n −MN
− 1).
The Fourier transform of H, H, is nonnegative andH(0) = 1.Using Poisson summation, the fact that χ(n) and e(n) haveabsolute value 1 and that |τ(χ)| =
√q for primitive
characters χ yields
|S∗(χ)| ≤ N√
q
∑k∈Z
(k ,q)=1
H(
kNq
).
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 10: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/10.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Proof of the Smoothed PV
Let H(t) = max{0,1− |t |}.
S∗(χ) =1
τ(χ)
q∑j=1
χ(j)∑n∈Z
e(nj/q)H(
n −MN
− 1).
The Fourier transform of H, H, is nonnegative andH(0) = 1.Using Poisson summation, the fact that χ(n) and e(n) haveabsolute value 1 and that |τ(χ)| =
√q for primitive
characters χ yields
|S∗(χ)| ≤ N√
q
∑k∈Z
(k ,q)=1
H(
kNq
).
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 11: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/11.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Proof
|S∗(χ)| ≤ N√
q
∑k∈Z
(k ,q)=1
H(
kNq
)
≤√
q
(∑k∈Z
Nq
H(
kNq
)− N
q
∑k∈Z
H(kN)
)
=√
q
(∑t∈Z
H(
qtN
)− N
qH(0)
)
=√
q − N√
q.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 12: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/12.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Corollaries
Corollary
Let χ be a primitive character to the modulus q > 1, let M,N be real numberswith 0 < N ≤ q and let m be a divisor of q such that 1 ≤ m ≤ q
N . Then
|S∗(χ)| ≤ φ(m)
m√
q.
Corollary
Let χ be a primitive character to the modulus q > 1, , then
|S∗(χ)| ≤ φ(q)q√
q + 2ω(q)−1 N√q.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 13: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/13.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Quick Application
Let D be a fundamental discriminant. Considerχ(n) =
(Dn
), where
(D.
)is the Kronecker symbol. Then χ is
a primitive Dirichlet character.Assume the least prime p such that χ(p) = −1 is greaterthan y for some y . Then
Sχ(N) =∑
n≤2N
χ(n)(
1−∣∣∣ nN− 1∣∣∣)
≥∑
n≤2N(n,D)=1
(1−
∣∣∣ nN− 1∣∣∣)− 2
∑y<p≤2Nχ(p)=−1
∑n≤ 2N
p(n,D)=1
(1−
∣∣∣npN− 1∣∣∣) .
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 14: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/14.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Quick Application
Let D be a fundamental discriminant. Considerχ(n) =
(Dn
), where
(D.
)is the Kronecker symbol. Then χ is
a primitive Dirichlet character.Assume the least prime p such that χ(p) = −1 is greaterthan y for some y . Then
Sχ(N) =∑
n≤2N
χ(n)(
1−∣∣∣ nN− 1∣∣∣)
≥∑
n≤2N(n,D)=1
(1−
∣∣∣ nN− 1∣∣∣)− 2
∑y<p≤2Nχ(p)=−1
∑n≤ 2N
p(n,D)=1
(1−
∣∣∣npN− 1∣∣∣) .
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 15: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/15.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Imprimitive Case
Corollary (Levin, Pomerance, Soundararajan)
Let χ be a primitive character to the modulus q > 1, then
|S∗(χ)| ≤ q3/2
N
{Nq
}(1−
{Nq
}). (1)
In particular, |S∗(χ)| < √q.
TheoremLet χ be a non-principal Dirichlet character to the modulus q > 1,then
|S∗(χ)| < 4√6√
q.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 16: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/16.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Lowerbound for the smoothed Pólya–Vinogradov
Theorem (ET)Let χ be a primitive character to the modulus q > 1, then
S∗(χ) ≥ 2π2√
q.
Therefore, the order of magnitude of S∗(χ) is√
q.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 17: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/17.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
More Work
Let q > 1 be an integer and let χ be a primitive Dirichletcharacter mod q. Let
Sχ(M,N) =∑
M≤n≤2N
χ(n)
(1−
∣∣∣∣n −MN
− 1∣∣∣∣) .
We’re interested in
A(q) = minχ
maxM,N|Sχ(M,N)| ,
andB(q) = max
χmaxM,N|Sχ(M,N)| .
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 18: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/18.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
From earlier theorems we know
A(q) >2π2 ,
andB(q) < 1.
Kamil Adamczewski wrote code to find A(q) and B(q) forsmall q (q ≤ 200).A(q) is around 0.45 in all modulus that have beencomputed. In those examples, B(q) is between 0.7 and0.8.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 19: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/19.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
From earlier theorems we know
A(q) >2π2 ,
andB(q) < 1.
Kamil Adamczewski wrote code to find A(q) and B(q) forsmall q (q ≤ 200).A(q) is around 0.45 in all modulus that have beencomputed. In those examples, B(q) is between 0.7 and0.8.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 20: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/20.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
From earlier theorems we know
A(q) >2π2 ,
andB(q) < 1.
Kamil Adamczewski wrote code to find A(q) and B(q) forsmall q (q ≤ 200).A(q) is around 0.45 in all modulus that have beencomputed. In those examples, B(q) is between 0.7 and0.8.
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality
![Page 21: The Smoothed Pólya Vinogradov Inequalitycampus.lakeforest.edu/trevino/IntegersSmoothedPolya...Pólya–Vinogradov was used to prove a conjecture of Mollin (Granville, Mollin and Williams)](https://reader034.vdocuments.mx/reader034/viewer/2022042419/5f362c9076d4f013973f365c/html5/thumbnails/21.jpg)
Pólya–VinogradovSmoothed Pólya–Vinogradov
Thank you!
Enrique Treviño The Smoothed Pólya–Vinogradov Inequality