Large values of short character sums
Alexander Kalmynin∗∗

National Research University Higher School of Economics, Moscow, Russia

Uniform Distribution Theory – 2018
5 October 2018

A. Kalmynin

Large values of short character sums

1. Introduction

Let χ be a Dirichlet character, i.e. completely multiplicative function,
which is periodic with some period q and satisfies a condition
χ(n) = 0 if (n, q) > 1.

A. Kalmynin

Large values of short character sums

1. Introduction

Let χ be a Dirichlet character, i.e. completely multiplicative function,
which is periodic with some period q and satisfies a condition
χ(n) = 0 if (n, q) > 1.
Let us also assume that χ is primitive i.e. it is not induced by anyother Dirichlet character.

A. Kalmynin

Large values of short character sums

1. Introduction

Let χ be a Dirichlet character, i.e. completely multiplicative function,
which is periodic with some period q and satisfies a condition
χ(n) = 0 if (n, q) > 1.
Let us also assume that χ is primitive i.e. it is not induced by anyother Dirichlet character. In number theory, the estimates for thequantityXχ(n)M <n≤N +M

are often very useful.

A. Kalmynin

Large values of short character sums

1. Introduction

Let χ be a Dirichlet character, i.e. completely multiplicative function,
which is periodic with some period q and satisfies a condition
χ(n) = 0 if (n, q) > 1.
Let us also assume that χ is primitive i.e. it is not induced by anyother Dirichlet character. In number theory, the estimates for thequantityXχ(n)M <n≤N +M

are often very useful. The Polya-Vinogradov inequality states that√sum over any interval is O( q ln q) and Burgess’s bound gives a
nontrivial estimate O(N p−c(ε) ) provided N ≥ p1/4+ε .

A. Kalmynin

Large values of short character sums

If we consider sums only over initial interval
X
χ(n)
n≤x

then there are also certain conditional results.

A. Kalmynin

Large values of short character sums

If we consider sums only over initial interval
X
χ(n)
n≤x

then there are also certain conditional results. If Generalized RiemannHypothesis is true, then for any ε > 0 we haveX√
χ(n)  xq εn≤x

A. Kalmynin

Large values of short character sums

If we consider sums only over initial intervalX
χ(n)
n≤x

then there are also certain conditional results. If Generalized RiemannHypothesis is true, then for any ε > 0 we haveX√χ(n)  xq εn≤x

and some heuristics also show that q ε can be replaced bys!!
ln q
exp O
ln ln qA. Kalmynin

Large values of short character sums

2. Quadratic characters

Lets restrict ourselves to the case when our character is quadratic (χ2is principal).

A. Kalmynin

Large values of short character sums

2. Quadratic characters

Lets restrict ourselves to the case when our character is quadratic (χ2is principal).
For example, if p is a prime number then one can consider thefollowing character

A. Kalmynin

Large values of short character sums

2. Quadratic characters

Lets restrict ourselves to the case when our character is quadratic (χ2is principal).
For example, if p is a prime number then one can consider thefollowing character  
+1 if n is quadratic residue mod pn= −1 if n is nonresiduep
0 if n is divisible by pMore generally, every primitive quadratic character can be written as
Dn where D is a fundamental discriminant.

A. Kalmynin

Large values of short character sums

It turns out that conditional bound that we mentioned before can besubstantially improved:
Theorem 1 (Granville, Soundararajan, 2001)
Let ω(q) be any function that tends to infinity for q → +∞. If
Generalized Riemann Hypothesis is true thenX
χ(n) = o(x)n≤x

for all primitive characters to the modulus q and x = (ln q)ω(q) .

A. Kalmynin

Large values of short character sums

It turns out that conditional bound that we mentioned before can besubstantially improved:
Theorem 1 (Granville, Soundararajan, 2001)
Let ω(q) be any function that tends to infinity for q → +∞. If
Generalized Riemann Hypothesis is true thenX
χ(n) = o(x)n≤xfor all primitive characters to the modulus q and x = (ln q)ω(q) .
Theorem 2 (Granville, Soundararajan, 2001)
For all sufficiently large q and any A > 0 there exists a fundamental
discriminant D with q ≤ |D| ≤ 2q such that the inequality
X D
xnn≤xholds, where x = ( 31 ln q)A .
A. Kalmynin

Large values of short character sums

3. The case of prime conductor

It is interesting to show that Theorem 2 holds for some smaller sets ofdiscriminants.

A. Kalmynin

Large values of short character sums

3. The case of prime conductor

It is interesting to show that Theorem 2 holds for some smaller sets ofdiscriminants.
Theorem 3 (Kalmynin, 2017)
Let A ≥ 1 be an arbitrarily large fixed number, x ≥ x0 (A) andy = (ln x)A . Then there exists a prime number p with x < p ≤ 2x such
thatX nypn≤yA. Kalmynin

Large values of short character sums

3. The case of prime conductor

It is interesting to show that Theorem 2 holds for some smaller sets ofdiscriminants.
Theorem 3 (Kalmynin, 2017)
Let A ≥ 1 be an arbitrarily large fixed number, x ≥ x0 (A) andy = (ln x)A . Then there exists a prime number p with x < p ≤ 2x suchthatX nypn≤ySo, sums of Legendre characters of length (ln p)A are not o((ln p)A ).

A. Kalmynin

Large values of short character sums

3. The case of prime conductor

It is interesting to show that Theorem 2 holds for some smaller sets ofdiscriminants.
Theorem 3 (Kalmynin, 2017)
Let A ≥ 1 be an arbitrarily large fixed number, x ≥ x0 (A) andy = (ln x)A . Then there exists a prime number p with x < p ≤ 2x such
thatX nypn≤ySo, sums of Legendre characters of length (ln p)A are not o((ln p)A ).
Proof of this result uses some properties of Siegel zeros of Dirichlet
L-functions.

A. Kalmynin

Large values of short character sums

4. Ideas of proof

Let us denote S(p, y) =Pn≤y np. One might try to compute theaverage value of S(p, y) over primes and hope to get something large,√but it turns out that it is much smaller than y (it is of order y bySiegel-Walfisz theorem).

A. Kalmynin

Large values of short character sums

4. Ideas of proof

Let us denote S(p, y) =Pn≤y np. One might try to compute theaverage value of S(p, y) over primes and hope to get something large,√but it turns out that it is much smaller than y (it is of order y bySiegel-Walfisz theorem).
So, our main idea is to introduce some weights wp that are very
biased towards large values of S(p, y).

A. Kalmynin

Large values of short character sums

4. Ideas of proofLet us denote S(p, y) =Pn≤y np. One might try to compute theaverage value of S(p, y) over primes and hope to get something large,√but it turns out that it is much smaller than y (it is of order y bySiegel-Walfisz theorem).
So, our main idea is to introduce some weights wp that are very
biased towards large values of S(p, y).
My choice of wp is as follows:

wp =Y q≤M   ( π(M )2
if for all primes q ≤ M we have pq = 1
q=1+p0 everywhere else

Here M = (ln x)1/3 .

A. Kalmynin

Large values of short character sums

So, the main goal is to prove that
PS(p, y)wp ln px<p≤2xPy
wp ln px<p≤2x

A. Kalmynin

Large values of short character sums

So, the main goal is to prove that
PS(p, y)wp ln px<p≤2xPy
wp ln px<p≤2x

Let us denote the numerator and denominator of this expression
S0 (x) and S1 (x, A).

A. Kalmynin

Large values of short character sums

So, the main goal is to prove that
PS(p, y)wp ln px<p≤2xPy
wp ln px<p≤2x

Let us denote the numerator and denominator of this expression
S0 (x) and S1 (x, A).
To provide an asymptotic formulas for S0 and S1 we need to deal withzeros of some Dirichlet L-functions
L(s, χ) =+∞X

χ(n)n−sn=1

A. Kalmynin

Large values of short character sums

We will use the following two classical theorems

A. Kalmynin

Large values of short character sums

We will use the following two classical theorems
Theorem 4For some constants √c and c1 > 0 and any nonprincipal character χ tothe modulus q ≤ ec1 ln x we haveXχ(p) ln p = −δ

p≤x



√
xβ+ O xe−c1 ln x .
βWhere δ = 1 if L(s, χ) has a real zero β with β > 1 − lncq and δ = 0otherwise.

A. Kalmynin

Large values of short character sums

We will use the following two classical theorems
Theorem 4
For some constants √c and c1 > 0 and any nonprincipal character χ tothe modulus q ≤ ec1 ln x we haveXχ(p) ln p = −δp≤x√xβ+ O xe−c1 ln x .
βWhere δ = 1 if L(s, χ) has a real zero β with β > 1 − lncq and δ = 0otherwise.
These exceptional real zeros are called Siegel zeros.

A. Kalmynin

Large values of short character sums

Theorem 5
Let χ1 and χ2 be two different real characters to the moduli q1 and q2 .
Let β1 and β2 be a real zeros of L(s, χ1 ) and L(s, χ2 ) repectively.
Then we havemax{β1 , β2 } ≤ 1 − pc2
max{q1 , q2 }andmin{β1 , β2 } ≤ 1 −c3ln(q1 q2 )So, Siegel zeros cannot be too large and they are quite rare.

A. Kalmynin

Large values of short character sums

Theorem 5Let χ1 and χ2 be two different real characters to the moduli q1 and q2 .
Let β1 and β2 be a real zeros of L(s, χ1 ) and L(s, χ2 ) repectively.
Then we havemax{β1 , β2 } ≤ 1 − pc2
max{q1 , q2 }andmin{β1 , β2 } ≤ 1 −c3ln(q1 q2 )So, Siegel zeros cannot be too large and they are quite rare.
 In
particular, one can show that a family of L-functions L(s, f· ),where f ≤ e2M is squarefree can have at most one really large
exceptional zero.

A. Kalmynin

Large values of short character sums

Theorem 5
Let χ1 and χ2 be two different real characters to the moduli q1 and q2 .
Let β1 and β2 be a real zeros of L(s, χ1 ) and L(s, χ2 ) repectively.
Then we havemax{β1 , β2 } ≤ 1 − pc2
max{q1 , q2 }andmin{β1 , β2 } ≤ 1 −c3ln(q1 q2 )So, Siegel zeros cannot be too large and they are quite rare.
 In
particular, one can show that a family of L-functions L(s, f· ),where f ≤ e2M is squarefree can have at most one really large
exceptional zero.
We denote the corresponding squarefree integer by f and the real zeroby βf .
A. Kalmynin

Large values of short character sums

Using Theorems 4 and 5 one can obtain the following asymptotic
formulas:S0 (x) ∼ x − δ1(2x)βf − xβfβf

where δ1 = 1 if exceptional modulus f exists and divides P (M ) and 0otherwise

A. Kalmynin

Large values of short character sumsUsing Theorems 4 and 5 one can obtain the following asymptotic
formulas:S0 (x) ∼ x − δ1(2x)βf − xβfβf

where δ1 = 1 if exceptional modulus f exists and divides P (M ) and 0otherwise andS1 (x, A) ∼ xrA,x (1) − δ2(2x)βf − xβfrA,x (f ),βfwhere δ2 = 1 if f exists and 0 if it is not. Here rA,x is defined by theformularA,x (c) = #{(a, d) : 1 ≤ a ≤ y, d | P (M ), ad = ct2 for some integer t}

A. Kalmynin

Large values of short character sums

To prove Theorem 3 we need the following lemma about rA,xLemma 6ln y
Let c be squarefree. Then rA,x (c)  y Mif c - P (M ) andrA,x (c) = rA,x (1)  y otherwise.

A. Kalmynin

Large values of short character sums

To prove Theorem 3 we need the following lemma about rA,xLemma 6ln y
Let c be squarefree. Then rA,x (c)  y Mif c - P (M ) andrA,x (c) = rA,x (1)  y otherwise.

Now, we consider the following three cases:

A. Kalmynin

Large values of short character sums

To prove Theorem 3 we need the following lemma about rA,xLemma 6ln y
Let c be squarefree. Then rA,x (c)  y Mif c - P (M ) andrA,x (c) = rA,x (1)  y otherwise.

Now, we consider the following three cases:
f doesn’t exist. Then δ1 = δ2 = 0 and we get
S0 (x) ∼ x
S1 (x, A) ∼ xrA,x (1)  xy,therefore we are done.

A. Kalmynin

Large values of short character sums

To prove Theorem 3 we need the following lemma about rA,xLemma 6ln y
Let c be squarefree. Then rA,x (c)  y Mif c - P (M ) andrA,x (c) = rA,x (1)  y otherwise.

Now, we consider the following three cases:
f doesn’t exist. Then δ1 = δ2 = 0 and we get
S0 (x) ∼ x
S1 (x, A) ∼ xrA,x (1)  xy,therefore we are done.
f exists but it doesn’t divide P (M ). Then δ1 = 0 but δ2 = 1 andwe have
S0 (x) ∼ x
S1 (x, A) ∼ xrA,x (1) −A. Kalmynin(2x)βf − xβfrA,x (f )βfLarge values of short character sums

Lemma 6 shows that rA,x (f ) is much smaller than rA,x (1), therefore
we getS1 (x, A) ∼ xrA,x (1)  xyas in the previous case.

A. Kalmynin

Large values of short character sums

Lemma 6 shows that rA,x (f ) is much smaller than rA,x (1), therefore
we getS1 (x, A) ∼ xrA,x (1)  xyas in the previous case.
f exists and divides P (M ). Then δ1 = δ2 = 1 and so we can get
no nice asymptotic formula for S0 and S1 . We only get thefollowing facts:S0 (x) ∼ x −(2x)βf − xβfβf

andS1 (x, A) ∼ xrA,x (1) −A. Kalmynin(2x)βf − xβfrA,x (f ).
βfLarge values of short character sums

In this final case Lemma 6 gives us rA,x (1) = rA,x (f ) and soasymptotics for our sums are spoiled in exactly the same way!

A. Kalmynin

Large values of short character sums

In this final case Lemma 6 gives us rA,x (1) = rA,x (f ) and so
asymptotics for our sums are spoiled in exactly the same way!
Therefore, these strange functions with βf cancel out and we finallyget
S1 (x, A)∼ rA,x (1)  y.
S0 (x)

A. Kalmynin

Large values of short character sums

In this final case Lemma 6 gives us rA,x (1) = rA,x (f ) and so
asymptotics for our sums are spoiled in exactly the same way!
Therefore, these strange functions with βf cancel out and we finallyget
S1 (x, A)∼ rA,x (1)  y.
S0 (x)
This concludes the proof.

A. Kalmynin

Large values of short character sums

Thank you for yourattention!

A. Kalmynin

Large values of short character sums