around the riemann hypothesis

Around the Riemann Hypothesis

Gautami Bhowmik

Universite de Lille , France.

13 December 2018.

Gautami Bhowmik mmeetspatlln

Building Blocks

[Aristotle, Book 13, Metaphysics] The incomposite is prior tothe composite.

Theorem (Fundamental Theorem of Arithmetic)

Every positive integer can be written uniquely as a product of primes.

There exist infinitely many primes.

With finitely many proves.

How do we count primes up to a given point?Are they regularly distributed?How do we generate them?How many primes are there of the form 4n + 1 or 4n + 3?How many pairs of primes are there of the form (p, p + 2)?

[Zagier, 1975] The prime numbers .. grow like weeds among thenatural numbers, seeming to obey no other law than that ofchance, and nobody can predict where the next one will sprout.The second fact ..states just the opposite: that the primenumbers exhibit stunning regularity, that there are lawsgoverning their behaviour, and that they obey these laws withalmost military precision.

Gautami Bhowmik mmeetspatlln

Building Blocks

[Aristotle, Book 13, Metaphysics] The incomposite is prior tothe composite.

Theorem (Fundamental Theorem of Arithmetic)

Every positive integer can be written uniquely as a product of primes.

There exist infinitely many primes. With finitely many proves.

How do we count primes up to a given point?Are they regularly distributed?How do we generate them?How many primes are there of the form 4n + 1 or 4n + 3?How many pairs of primes are there of the form (p, p + 2)?

[Zagier, 1975] The prime numbers .. grow like weeds among thenatural numbers, seeming to obey no other law than that ofchance, and nobody can predict where the next one will sprout.The second fact ..states just the opposite: that the primenumbers exhibit stunning regularity, that there are lawsgoverning their behaviour, and that they obey these laws withalmost military precision.

Gautami Bhowmik mmeetspatlln

Building Blocks

[Aristotle, Book 13, Metaphysics] The incomposite is prior tothe composite.

Theorem (Fundamental Theorem of Arithmetic)

Every positive integer can be written uniquely as a product of primes.

There exist infinitely many primes. With finitely many proves.

How do we count primes up to a given point?Are they regularly distributed?How do we generate them?How many primes are there of the form 4n + 1 or 4n + 3?How many pairs of primes are there of the form (p, p + 2)?

[Zagier, 1975] The prime numbers .. grow like weeds among thenatural numbers, seeming to obey no other law than that ofchance, and nobody can predict where the next one will sprout.The second fact ..states just the opposite: that the primenumbers exhibit stunning regularity, that there are lawsgoverning their behaviour, and that they obey these laws withalmost military precision.

Gautami Bhowmik mmeetspatlln

Gautami Bhowmik mmeetspatlln

Gauss

Gauss predicted the density of primesupto x as 1

log x probably around 1792 .

Letter to Encke, 24th December 1849.

Gautami Bhowmik mmeetspatlln

Gauss

Gauss predicted the density of primesupto x as 1

log x probably around 1792 .

Letter to Encke, 24th December 1849.

Gautami Bhowmik mmeetspatlln

Logarithmic Integral

Gauss prediction :

π(x) ∼ Li(x) =

∫ x

2

dt

log t.

Remarkably good approximation with a ’square-root error’ .Difference in less than half the digits.Tchebychev ( 1848) First study of π(x) by analytic methods.

Gautami Bhowmik mmeetspatlln

Logarithmic Integral

Gauss prediction :

π(x) ∼ Li(x) =

∫ x

2

dt

log t.

Remarkably good approximation with a ’square-root error’ .Difference in less than half the digits.

Tchebychev ( 1848) First study of π(x) by analytic methods.

Gautami Bhowmik mmeetspatlln

Logarithmic Integral

Gauss prediction :

π(x) ∼ Li(x) =

∫ x

2

dt

log t.

Remarkably good approximation with a ’square-root error’ .Difference in less than half the digits.Tchebychev ( 1848) First study of π(x) by analytic methods.

Gautami Bhowmik mmeetspatlln

Discrete to Continuous : Contour Integrals

Discrete Problems of Number Theory → Complex Analysis

Example (Goldbach)

To find a− b = 0, use exponential function around the circle∫ 1

0

e2πintdt = { 1 if n = 00 otherwise.

To find p + q = n,

∑p,q≤n

∫ 1

0

e2πi(p+q−n)tdt =

∫ 1

0

e−2πint(∑p≤n

e2πipt)2dt.

To detect when x/n > 1 ; z = ey ,

Example (Perron)

1

2πi

∫σ

z s

sds = {

0 , 0 < z < 11/2 , z = 1

1 , z > 1.

Gautami Bhowmik mmeetspatlln

Discrete to Continuous : Contour Integrals

Discrete Problems of Number Theory → Complex Analysis

Example (Goldbach)

To find a− b = 0, use exponential function around the circle∫ 1

0

e2πintdt = { 1 if n = 00 otherwise.

To find p + q = n,

∑p,q≤n

∫ 1

0

e2πi(p+q−n)tdt =

∫ 1

0

e−2πint(∑p≤n

e2πipt)2dt.

To detect when x/n > 1 ; z = ey ,

Example (Perron)

1

2πi

∫σ

z s

sds = {

0 , 0 < z < 11/2 , z = 1

1 , z > 1.Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

Riemann

Counting primes using theory of complex functions.The Riemann Zeta Function (1859)

ζ(s) =∞∑n=1

1

ns, R(s) > 1.

ζ(s) =∏p

1

1− p−s.

Functional Equationζ(s) = ?ζ(1− s)

? = 2sπs−1 sin(πs/2)Γ(1− s)

ζ(s) can be continued meromorphically to the whole complex plane.

Proposition (Explicit Formula)∑pm≤x

log p = x −∑

ρ:ζ(ρ)=0

xρ

ρ− ζ ′(0)

ζ(0).

Gautami Bhowmik mmeetspatlln

RH

Conjecture (RH 1859)

All non-trivial zeros of the Riemann Zeta Function lie on the lineR(s) = 1/2.

Clay Institute Millenium PrizeGeorge Clooney earned 239 million dollars

between June 1, 2017 and June 1, 2018 (Forbes website).

Gautami Bhowmik mmeetspatlln

RH

Conjecture (RH 1859)

All non-trivial zeros of the Riemann Zeta Function lie on the lineR(s) = 1/2.

Clay Institute Millenium Prize

George Clooney earned 239 million dollarsbetween June 1, 2017 and June 1, 2018 (Forbes website).

Gautami Bhowmik mmeetspatlln

RH

Conjecture (RH 1859)

All non-trivial zeros of the Riemann Zeta Function lie on the lineR(s) = 1/2.

Clay Institute Millenium PrizeGeorge Clooney earned 239 million dollars

between June 1, 2017 and June 1, 2018 (Forbes website).

Gautami Bhowmik mmeetspatlln

PNT

Riemann gave a programme but could not actually prove the primenumber theorem.

Proposition

ζ(1 + it) 6= 0.

Theorem (PNT, de la Vallee Poussin, Hadamard 1896 )

π(x) ∼ x

log x

as x →∞.

Proposition (RH-PNT)

The RH is equivalent to |Li(x)− π(x)| ≤ x1/2 log(x) for all x ≥ 3.

Gautami Bhowmik mmeetspatlln

PNT

Riemann gave a programme but could not actually prove the primenumber theorem.

Proposition

ζ(1 + it) 6= 0.

Theorem (PNT, de la Vallee Poussin, Hadamard 1896 )

π(x) ∼ x

log x

as x →∞.

Proposition (RH-PNT)

The RH is equivalent to |Li(x)− π(x)| ≤ x1/2 log(x) for all x ≥ 3.

Gautami Bhowmik mmeetspatlln

PNT

Riemann gave a programme but could not actually prove the primenumber theorem.

Proposition

ζ(1 + it) 6= 0.

Theorem (PNT, de la Vallee Poussin, Hadamard 1896 )

π(x) ∼ x

log x

as x →∞.

Proposition (RH-PNT)

The RH is equivalent to |Li(x)− π(x)| ≤ x1/2 log(x) for all x ≥ 3.

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...

Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)

Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)

Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)

Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjecture

Quantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)

Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Consequences

Failure of RH would cause havoc in the distribution of primenumbers.

GRH : RH for a family of L functions

Dirichlet, Dedekind, Hasse-Weil...Automorphic (Godement -Jacquet (1976)).

Under GRH

Ternary Goldbach (Vinogradov 1954)Primality testing (Miller, AKS)Elliptic Curves (Serre 1981)Artin’s primitive root conjectureQuantum Chaos (Waldspurger 1981)Ternary Quadratic Forms (Ellenberg-Venkatesh 2008)

Proved for algebraic varieties over finite fields (Deligne 1974).

Gautami Bhowmik mmeetspatlln

Tools

Numerical Verification for 10 trillions of zeros.

Non-commutative geometry (Alain Connes)Spectral Interpretation : Hilbert and PolyaRandom MethodsZeros of ζ(s) < −−− > Eigenvalues of random matrices

↑||

Energy levels in quantum systems

Will physics bring the answer ????

Gautami Bhowmik mmeetspatlln

Tools

Numerical Verification for 10 trillions of zeros.Non-commutative geometry (Alain Connes)

Spectral Interpretation : Hilbert and PolyaRandom MethodsZeros of ζ(s) < −−− > Eigenvalues of random matrices

↑||

Energy levels in quantum systems

Will physics bring the answer ????

Gautami Bhowmik mmeetspatlln

Tools

Numerical Verification for 10 trillions of zeros.Non-commutative geometry (Alain Connes)Spectral Interpretation : Hilbert and PolyaRandom MethodsZeros of ζ(s) < −−− > Eigenvalues of random matrices

↑||

Energy levels in quantum systems

Will physics bring the answer ????

Gautami Bhowmik mmeetspatlln

Bedank

Dames en heren, ik dank u allen voor uw andacht.Merci de votre patience.

Gautami Bhowmik mmeetspatlln