Riemann's Memoir and the Related - BYU Mathlzhao/Presentations/riemann.pdf · Let N(T) be the number of zeros ˆof (s) with 0

OutlinePrime Numbers

Riemann zeta-functionReasons for belief and disbelief

References

Riemann’s Memoir and the Related

Liangyi Zhao

Division of Mathematical SciencesSchool of Physical and Mathematical Sciences

Nanyang Technological UniversitySingapore

18 November 2009

Liangyi Zhao Riemann’s Memoir and the Related



References

1 Prime NumbersInfinitude by EuclidInfinitude by EulerPrime Number Theorem

2 Riemann zeta-functionDefinitionRiemann’s MemoirThe Riemann HypothesisThe Explicit FormulaProof of PNT

3 Reasons for belief and disbeliefReasons for BeliefThe Devil’s AdvocateLehmer’s Phenonemon

4 References




References

Infinitude by EuclidInfinitude by EulerPrime Number Theorem

Infinitude by Euclid of Alexandria circa 22 B. C.

A prime number is a natural number greater than one that isdivisible only by one and itself. Let P henceforth denote theset of prime numbers.

The fundamental theorem of arithmetics (FTA) states thatevery natural number can be uniquely, up to re-ordering thefactors, written as a product of primes.

Consequently, since n! + 1 has no divisor between 2 and nthere exists a prime (dividing n! + 1) greater than n.

Therefore, there must be infinitely many prime numbers.




References


Infinitude by Euler in 1737

For n > 1, we have

log n =

∫ n

1

1

xdx ≤

n∑k=1

1

k≤∏p≤np∈P

(1 +

1

p+

1

p2+ · · ·

)≤∏p≤np∈P

1

1− 1p

.

Upon taking the logarithm of both side of the above, we have

log log n ≤ −∑p≤np∈P

log(1− p−1) =∑p≤np∈P

∞∑l=1

1

lpl,

where we have used the Taylor series

log(1− x) = −∞∑l=1

x l

l, for |x | < 1.




References



Now we have

log log n ≤∑p≤np∈P

∞∑l=1

1

lpl=∑p≤np∈P

1

p+∑p≤np∈P

∞∑l=2

1

lpl.

We can estimate the last sum above in the following way.

∑p≤np∈P

∞∑l=2

1

lpl≤

n∑m=2

∞∑l=2

m−l =n∑

m=2

1

m2(1−m−1)≤∞∑

m=2

1

m(m − 1).

Note that the last sum converges and the limit is 1.




References



Therefore, we must have,

log log n ≤∑p≤np∈P

1

p+ θ, with |θ| ≤ 1.

This means that as n tends to infinity,∑p≤np∈P

1

p→∞.

Hence there must be infinitely many prime numbers as finitesums converge.




References


How infinite are the prime numbers?

Let π(x) denote the number of primes not exceeding x .

Modifying Euclid’s proof would give π(x) ≥ log log x .

Euler’s proof would suggest that π(x) should be much larger.

It was first conjectured by Legendre that the ratio of π(x) andx

log x is 1 as x →∞. Hence π(x) is well-approximated by xlog x

if x is large.

Gauss wrote in 1849 that he reached the conclusion of theconjecture at the age of 15 in 1792, although what hebelieved was that π(x) is well-approximated by

li(x) =

∫ x

2

1

log tdt.




References


How infinite are the prime numbers?

It can be shown, using integration by parts, that

li(x) =x

log x+

x

log2 x[1 + o(1)], as x →∞.

It was in 1896 that Hadamard and de la Vallee Poussin provedthe conjecture independently.

Theorem (Prime Number Theorem)

There exists a constant c > 0, effectively computable such that forx ≥ 2

π(x) = li(x) + O(

x exp(−c√

log x))

,

where the implied constant is absolute.




References


How good is this approximation?

x π(x) [li(x)− π(x)]

108 5,761,455 754

109 50,847,534 1,701

1010 455,052,511 3,104

1011 4,118,054,813 11,588

1012 37,607,912,018 38,263

1013 346,065,536,839 108,971

1014 3,204,941,750,802 314,890

1015 29,844,570,422,669 1,052,619

1016 279,238,341,033,925 3,214,632




References

DefinitionRiemann’s MemoirThe Riemann HypothesisThe Explicit FormulaProof of PNT

Riemann zeta-function

Let

ζ(s) =∞∑

n=1

1

ns=∏p∈P

(1− p−s

)−1, for<s > 1.

The infinite sum and product are the same by FTA.

It is much more natural to count the primes with a weight.

Set

Λ(n) =

{log p; if n = pk ,

0, otherwise.

The prime number theorem is equivalent to

ψ(x) :=∑n≤x

Λ(n) = x + O(

x exp(−c√

log x))

.




References


Riemann zeta-function

The reason for counting primes this way is that it brings ζ(s)into the picture.

−ζ′(s)

ζ(s)= − d

dslog ζ(s) =

∑p∈P

log pp−s

1− p−s

=∑p∈P

log p(p−s + p−2s + · · ·

)=∞∑

n=1

Λ(n)

ns, for <s > 1.

Euler was the first to study ζ(s), only considering the functionas that of a real variable.

Riemann introduced complex analysis to the investigation and,in doing so, set a new direction for mathematics.




References


Riemann’s Memoir

In 1859, Georg Friedrich Bernhard Riemann, a newly electedmember of the Berlin Academy of Sciences reporting on hismost recent research, sent an article titled Ueber die Anzahlder Primzahlen unter einer gegebenen Grosse (On the Numberof Primes Less than a Given Magnitude) to the academy.

Considering that it was his only paper in the theory ofnumbers and changed the direction of mathematical researchin very significant ways, it is now appropriately and betterknown as “Riemann’s Memoir.”

The reason for this dubbing is perhaps also partly due to thefact that Riemann fell seriously ill two years later and passedon in 1866, almost two months before his fortieth birthday.




References


Riemann’s Theorems

ζ(s) can be continued meromorphically to the whole of Cwith only one pole at s = 1 of residue 1.

ζ(s) =s

s − 1− s

∫ ∞1{x}x−s−1dx , for <s > 0.

ζ(s) satisfies the functional equation

π−s2 Γ( s

2

)ζ(s) = π−

1−s2 Γ

(1− s

2

)ζ(1− s).

Hence the values of ζ(s) for <s < 0 can be obtained fromthose for <s > 1. In particular ζ(s) = 0 if s is a negative eveninteger since Γ(s/2) has a pole there. These are known as thetrivial zeros.




References


Riemann’s Conjectures

ζ(s) has infinitely many zeros ρ with 0 < <ρ < 1 (the criticalstrip), symmetric about the lines =s = 0 and <s = 1

2 . Thesymmetry follows from Riemann’s theorems. The infinitude isa consequence of the next conjecture.

Let N(T ) be the number of zeros ρ of ζ(s) with 0 ≤ <ρ ≤ 1and 0 ≤ =ρ ≤ T . Then

N(T ) =T

2πlog

T

2π− T

2π+ O(log T ).

This is known as the Riemann-von Mangoldt formula whichwas first proved in 1895 by von Mangoldt in a less satisfyingform and fully in 1905.




References


Riemann’s Conjectures in his Memoir

The function

ξ(s) =1

2s(s − 1)π−

s2 Γ( s

2

)ζ(s)

is entire and satisfies a product formula

ξ(s) = exp(A + Bs)∏ρ

(1− s

ρ

)exp

(s

ρ

),

where A,B ∈ R and ρ runs over the zeros of ζ(s) in thecritical strip. This was proved by Hadamard in 1893.There is an explicit formula for π(x)− li(x), or equivalently

ψ(x)− x = −∑ρ

xρ

ρ− ζ ′(0)

ζ(0)− 1

2log

(1− 1

x2

)which was proved in 1895 by von Mangoldt.




References


The Riemann hypothesis

The only unsolved conjecture in Riemann’s Memoir is theRiemann hypothesis(RH). It states that all the non-trivialzeros of ζ(s) have real part 1

2 .

It is not clear what led Riemann to this conjecture, or any ofthe ones mentioned above. But it seems that Riemann knew alot more about ζ(s) than is apparent in the published memoir.

Riemann was cautious in his memoir, using the words ”verylikely”(”sehr wahrscheinlich”) in connection with RH.

”One would of course like to have a rigorous proof of this[RH], but I have put aside the search for such a proof aftersome fleeting vain attempts because it is not necessary for theimmediate objective of my investigation.”




References


The Riemann’s hypothesis

Hardy in 1914 showed that infinitely many zeros lie on thecritical line, the line with <s = 1

2 .

A more precise statement of this was later given by Hardy andLittlewood, that the number of zeros on the critical line withimaginary parts in [0,T ] is greater than cT for some c > 0.

In 1942, A. Selberg proved that a positive proportion ofnon-trivial zeros have real part 1

2 .

In 1974, Levinson showed that at least one third of all zerosare at the correct place.

Conrey improved this proportion to 40% in 1991.

No counter example to RH has been found in the first1.5× 109 zeros of ζ(s) in the critical strip.




References


Proof of the Explicit Formula

If c > 0, then.

1

2πi

∫(c)

y s ds

s=

0, if 0 < y < 1,

1/2, if y = 1,1, if y > 1,

where

∫(c)

=

∫<s=c

.

Therefore, if c > 1, ψ(x) =∑

n≤x Λ(n)

≈ 1

2πi

∞∑n=1

Λ(n)

∫(c)

(x

n

)s ds

s=−1

2πi

∫(c)

ζ ′(s)

ζ(s)

x s

sds.

Move the contour to the left and collect all residues, we get

ψ(x)− x = −∑ρ

xρ

ρ− ζ ′(0)

ζ(0)− 1

2log

(1− 1

x2

).




References


Proof of the Explicit Formula

From the explicit formula below, it is important, in order toknow the size of the sum over ρ; we need an upper bound forthe real parts of these ρ’s.

ψ(x)− x = −∑ρ

xρ

ρ− ζ ′(0)

ζ(0)− 1

2log

(1− 1

x2

)The best case scenario, because of the symmetry of the ρ’s, is<ρ = 1/2 for all ρ.




References


Proof of PNT

We use a truncated version of the discrete integral.

This gives

ψ(x)− x =∑|=ρ|<T

xρ

ρ+ error terms.

A zero-free region enables us to get the error term in PNT.

There is c > 0 such that if ρ = β + iγ with β > 0 andζ(ρ) = 0, then β < 1− c/ log γ.

Partial summation allows us to go from ψ(x) to π(x).

RH would imply the error term in PNT can be taken to be

O(√

x log x).




References

Reasons for BeliefThe Devil’s AdvocateLehmer’s Phenonemon

Reasons for Belief

A sanguine disposition: mathematics should be beautiful, thedistribution of primes should be as ”regular” as possible.

One can infer from that Euler product that for <s > 1,ζ−1(s) =

∑∞n=1 µ(n)n−s where µ(n) is zero if n is not

square-free and (−1)k if n is the product of k distinct primes.

RH is equivalent to∑n≤x

µ(n) = Oε

(x1/2+ε

).

µ(n) appears to take on ±1 fairly randomly and hence onemay expect the sum above to have a lot of cancellations,giving the desired estimate.




References


Reasons for Belief

A VERY large number of zeros have been computed and nocounter example to RH has been found.

A large proportion can be proved to be very close to the line,i. e. zero density theorems.

RH-like statements are known to hold for other functionssimilar to ζ(s), e. g. zeta functions for function fields (Weil),algebraic varieties over finite fields (Deligne).

Certain average versions of RH have been proved, e. g.Bombieri-Vinogradov theorem,




References


Playing the Devil’s Advocate

RH-like statements are known NOT to hold for somefunctions with properties similar to those of ζ(s), e.g.Davenport-Heilbronn zeta-function, Epstein zeta-function.

The hitherto also unresolved Lindelof hypothesis (LH), whichis a consequence of RH, asserts

ζ(1/2 + it) = Oε (tε) for any ε > 0.

There are heuristics that would lead to the falsehood LH.

True behaviors of ζ(s) may not be apparent for s with smallimaginary part. RH implies that the error term in theRiemann-von Mangoldt formula may be taken to beO (log x/ log log x), but the actual error (known to beunbounded) has never been seen to be much larger than 3.




References


Lehmer’s Phenomenon

We consider a certain real-valued Z (t) with|Z (t)| = |ζ(1/2 + it)|.RH implies that Z (t) has no positive local minimum ornegative local maximum for t > 1000.

The term ”Lehmer’s phenomenon” refers to a behavior ofZ (t) that its graph sometimes barely crosses the t-axis, analmost counter example of RH.

This shows the delicacy of ζ(s) and ”must give pause to eventhe most convinced believer in the Riemann hypothesis.”




References

Further Readings

H. M. Edwards, Riemann’s Zeta Function

A. Ivic, The Riemann Zeta-function: Theory and Applications

S. J. Patterson, An Introduction to the Theory of theRiemann Zeta-function.

E. C. Titchmarsh, The Theory of the Riemann Zeta-function


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Riemann's Memoir and the Related - BYU Mathlzhao/Presentations/riemann.pdf · Let N(T) be the number of zeros ˆof (s) with 0