littlewood and number theory

789RESONANCE September 2013

GENERAL ARTICLE

Littlewood and Number Theory

M Ram Murty

Keywords

Circle method, zeroes of Ri-

emann zeta function, Diophan-

tine approximation, Waring prob-

lem, Goldbach conjecture.

M Ram Murty, a

Canadian mathematician,

is currently Queen's

Research Chair in

Mathematics and

Philosophy at the Queen's

University, Canada. He is

also an adjunct professor

at several institutions in

India (TIFR in Mumbai,

IMSc in Chennai and HRI

in Allahabad.) He also

teaches Indian philosophy

at Queen's University and

has recently published a

book titled Indian

Philosophy and has written

many popular monographs

for students of

mathematics.

J E Littlewood (18851977) was a British math-ematician well known for his joint work withG H Hardy on Warings problem and the devel-opment of the circle method. In the rst quarterof the 20th century, they created a school of anal-ysis considered the best in the world. Littlewoodrmly believed that research should be oset bya certain amount of teaching. In this exposition,we will highlight several notable results obtainedby Littlewood in the area of number theory.

1. Introduction

John Edensor Littlewood was one of the most well-knownmathematicians during the rst part of the 20th century.Most notable among his doctoral students were A O LAtkin, Sarvadaman Chowla, Harold Davenport, A E In-gham and H P F Swinnerton-Dyer. For ocial reasons,Srinivasa Ramanujan is also considered as one of Little-woods students.

In his obituary on Littlewood, Burkill [1] observes thataround 1900, there was no British school of analysis wor-thy of the name and it is due to G H Hardy and J ELittlewood that England owes a debt of gratitude forbuilding by 1930 a school of analysis second to none inthe world.

2. Prime Numbers

Littlewoods rst result in prime number theory con-cerned the error term in the prime counting function.If (x) denotes the number of primes up to x, then theprime number theorem tells us that

(x) lix,

790 RESONANCE September 2013

GENERAL ARTICLE

Littlewood proved the

surprising fact that the

differenceS(x) li (x)between the prime

counting function S(x)and the logarithmic

integral li (x) changes

sign infinitely often.

where

lix =

x2

dt

log t

is called the logarithmic integral. Here, the symbolA(x) B(x) means that A(x)/B(x) tends to 1 as xtends to innity. Numerical evidence seemed to suggestthat

(x) < li x,

but Littlewood showed that this was incorrect. In animportant and substantial paper, he [2] showed that

(x) lixchanges sign innitely often. More precisely, he provedthat there are positive constants c1, c2 such that

(x) li x > c1x1/2 log log log x

log x

for innitely many x tending to innity, and

(x) lix < c2x1/2 log log log x

log x

for innitely many x tending to innity. One usuallywrites this fact as

(x) lix = (

x1/2 log log log x

log x

).

3. The Approximate Functional Equation of theZeta Function

However, what distinguishes Littlewood in the historyof mathematics is his prolic and productive collabora-tion with G H Hardy. Together, they tackled some ofthe fundamental questions regarding the Riemann zetafunction and related questions. As is well known, theRiemann zeta function is dened by the series

(s) =

n=1

1

ns, (s) > 1.

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Hardy and

Littlewood showed

that the Riemann

zeta function has

infinitely many

zeroes on the critical

line (s) = 1/2.

By virtue of the unique factorization of the natural num-bers one can write this as an innite product over primenumbers:

(s) =p

(1 1

ps

)1,

for (s) > 1. In his 1860 paper, Riemann derived theanalytic continuation and functional equation for (s).More precisely, he showed that

s/2(s2

)(s) = (1s)/2

(1 s2

)(1 s).

Here, the Euler gamma function is dened by (z) =0

xz1exdx for (z) > 0. Beautiful as the abovefunctional equation is, it is impractical to use in manyquestions of analytic number theory. In 1921, Hardyand Littlewood [3] derived the following approximatefunctional equation:

(s) =nx

ns + (s)ny

n(1s) + O(x)

+ O(|t|1/2y1) ,

where s = + it, t real, and 2xy = |t|, with

(s) = 2(2)s1(1 s) sin s2

.

Here, we write for non-negative b(x), a(x) = O(b(x)) ifthere is a constant K such that |a(x)| Kb(x) for allx. Using this, they were able to show that the num-ber of zeros of (s) with |(s)| T that lie on the line(s) = 1/2 is greater than cT for some positive con-stant c. (Of course, the celebrated Riemann hypothesissays that all of the non-trivial zeros of (s) lie on theline (s) = 1/2.) The HardyLittlewood theorem wasimproved by Selberg [4] in 1941 who showed that a pos-itive proportion of the non-trivial zeros of (s) lie on thecritical line. In 1974, Levinson [5] showed that one-third


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The possible

exceptions to

Littlewood's

conjecture from

around 1930 on

simultaneous

approximation of two

real numbers by

rationals with the

same denominator,

have been proved in

2006 to have zero

Hausdorff dimension.

of the zeros lie on the critical line and in 1989, Conrey[6] improved this to show that two-fths of the zeroslie on the critical line. The seminal work of Hardy andLittlewood initiated these subsequent developments.

Another important application of the approximate func-tional equation is the beautiful asymptotic formula:

T0

|(12+ it)|2dt T log T.

In 1934, E.C. Titchmarsh [7] improved this to show

T0

|(12+ it)|2dt = T (logT log 2+21)+O(T +),

where is Eulers constant and = 5/12. In his TIFRdoctoral thesis, R Balasubramanian [8] improved thisfurther and showed that one can take = 1/3.

4. Littlewoods Problem on Diophantine Ap-proximation

For any real number x, let ||x|| denote the distance ofx to the nearest integer. For any pair of real numbers, , Littlewood conjectured that

lim infn

n||n||||n|| = 0.

Another way to formulate this is that for any > 0, theinequality

|x(x y)(x z)| < ,

can be solved with x = 0 and x, y, z integers. The func-tion L(x, y, z) = x(x y)(x z) is a product of threelinear forms which admits a two-dimensional torus asa group of automorphisms. Recently, this conjecturehas received considerable attention because it ts into alarger framework of ergodic theory and Lie theory. Us-ing new methods from these theories, the authors in [9]


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showed that the set of exceptions to Littlewoods con-jecture has Hausdor dimension zero.

5. Warings Problem and the Development ofthe Circle Method

The circle method originates with Ramanujan as canbe seen by entries of his rst letter to G H Hardy in1913 (see for example, (1.14) on page 8 of [10] as well asSelberg [11]). After his arrival in England, Hardy andRamanujan applied the method to derive the asymptoticformula for the partition function. After Ramanujanspremature demise, Hardy and Littlewood developed thismethod further and applied it rst to tackle Waringsproblem which we now explain.

A celebrated theorem of Lagrange states that every nat-ural number can be written as the sum of four squarenumbers. Using elementary number theory, one canshow that numbers of the form 8n + 7 cannot be writ-ten as the sum of three or fewer squares. Indeed, anysquare mod 8 is congruent to 0, 1 or 4 and it is impossi-ble to choose three of these numbers (with replacement)to obtain 7 (mod 8). In 1770, Edmund Waring asked ifgenerally, for any value of k, there is a number g(k) suchthat every number can be written as a sum of g(k) num-ber of k-th powers and such that nothing smaller thang(k) works. Thus, Lagranges theorem states g(2) = 4.Waring conjectured further that g(3) = 9, g(4) = 19and so on. That g(k) exists was shown by Hilbert in1909. However, obtaining an explicit formula for g(k)was another matter. Hilberts method, though elemen-tary in the technical sense of the word, employed com-plicated algebraic identities and it was dicult to obtainany further information regarding g(k) from his method.Moreover, if n can be written as a sum of r k-th powers,is it possible to obtain an asymptotic formula for thenumber of such representations? The answers to thesequestions were not accessible by Hilberts method. They


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were however obtainable via the powerful circle methodas developed by Hardy and Littlewood in their series ofpapers on the subject (see for instance, [12]).

It is easy to see that to write the number 3k1 as a sumof k-th powers, we can only use 1 and 2k. To use theminimum number of k-th powers, we use ([(3/2)k] 1)2ks and the remaining 2k1 summands are all 1 so that

g(k) 2k + [(3/2)k] 2.It seems that J A Euler, the son of the famous LeonardEuler, was the rst to see this. Thus, g(k) 2k +[(3/2)k]2 for all values of k. In 1933, Pillai (see p. xiiiof [13]) proved that g(k) = 2k + [(3/2)k] 2 providedthe following inequality holds:

(3/2)k [(3/2)k] 1 (3/4)k.In 1957, Mahler [14] showed that the inequality is truefor k suciently large, but was unable to determine howlarge, since his proof was ineective. This is still an openproblem today.

However, the powerful circle method enabled Hardy andLittlewood to obtain asymptotic formulas for rk,s(n), thenumber of ways of writing n as a sum of s k-th powersprovided s > 2k. More precisely, they showed that fors > 2k,

rk,s(n) (1 + 1/k)s

(s/k)Sk,s(n)n

s/k1,

where Sk,s(n) 1. The factor Sk,s(n) is called the sin-gular series and can be written as a product over primes

Sk,s(n) =p

A(p),

where A(p) can be interpreted as a p-adic density in thefollowing sense. For each natural number q, let M(q) bethe number of solutions of the congruence

xk1 + + xks n(mod q).

The circle method is

powerful enough to

obtain asymptotic

expressions for the

number of represen-

tations of a positive

integer as the sum of

a (fixed) number of

k-th powers.


GENERAL ARTICLE

Although the circle

method gives

expressions for the

number of ways of

decomposing any

large enough odd

number as a sum of

three primes, it is not

sufficient to prove

Goldbachs conjecture

that every even

number !2 is the sumof two primes.

Then,A(p) = lim

rM(pr)pr(s1).

Warings problem is still an area of active research eventoday as is outlined in [15].

6. The Circle Method and Goldbachs Conjec-ture

After the successful series of papers on the Waring prob-lem, Hardy and Littlewood realized that the method isapplicable to study other questions such as the Gold-bachs conjecture that every even number can be writtenas the sum of two primes and that every odd number canbe written as the sum of three primes. Indeed, if rk(n)denotes the number of ways of writing n as a sum of kprimes. Hardy and Littlewood [16] showed that for oddn, we have

r3(n) S3(n) n2

2 log3 n, n,

where

S3(n) =p

(1 +

p(n)

(p 1)3)

with p(n) = (p 1) if p|n and 1 otherwise, providedthe generalized Riemann hypothesis (GRH) is true. Thishypothesis is the generalization of the classical Riemannhypothesis for the Riemann zeta function and pertainsto Dirichlet L-series, L(s, ) with a Dirichlet charac-ter. In 1937, Vinogradov derived some new estimatesfor exponential sums of the form

px

e2ip ,

that enabled him to remove the use of the GRH andso the HardyLittlewood theorem is now unconditional.For the binary Goldbach problem however, the methoddoes not work. Hardy and Littlewood were able to


GENERAL ARTICLE

Hardy and Littlewood

conjectured that the

prime counting

function S(x) satisfiesS(x + y) d S(x) + S(y)

for all x, y >1; it is

expected to be false.

use the method to conjecture an asymptotic formula forr2(n). Applying the same heuristic reasoning, they con-jectured that the number of twin primes p x (that isprimes p such that p + 2 is also prime) is asymptotic to

S2(2)x

log2 x

as x tends to innity. In the same paper [16], theyused the circle method to formulate a general conjec-ture about k-tuplets of primes. This conjecture saysthe following. Let H = {h1, ..., hk} be a set of distinctnatural numbers. For each prime p, let p(H) be thesize of the image of H (mod p). The HardyLittlewoodconjecture is that the number of n x such that

n + h1, ..., n+ hk

are all prime is asymptotic to

S(H)x

logk x,

where S(H) =p

(1 p(H)

p

)(1 1

p

)k,

where the product is over primes p. Surprisingly, in thesame paper they conjectured that

(x+ y) (x) + (y),

for all x, y 2. In 1974, Hensley and Richards [17]showed that this latter conjecture is incompatible withthe k-tuplets conjecture and so one of them must bewrong. It is generally believed that it is the latter con-jecture which is false, though no one has yet proved thisis the case. The motivation for making the latter conjec-ture is based on the intuition that there are more primesin the initial interval [1, x] of length x than there are inthe later interval [y, y + x] of length x. This example


GENERAL ARTICLE

Usually, Littlewood

would write the

penultimate draft

of their joint paper

and Hardy would

write the flowery

prose of which he

was a master.

shows how sometimes, our intuition may be wrong, evenfor the best of minds.

7. Concluding Remarks

After the great collaborative work of Hardy and Ra-manujan, the joint work of Hardy and Littlewood hadacquired the status of epic collaboration in the annals ofmathematics. Legend has it that they had four axiomsfor this collaborative work. The rst was that when onewrote to the other it did not matter whether what wassaid was right or wrong, for otherwise, they could notwrite freely and openly as they pleased. The second ax-iom was that when one had received a message from theother, he was under no obligation to either read it or torespond to it. The third was that it did not matter ifthey both thought about the same detail. Finally, thefourth axiom was that it did not matter if one of themhad contributed nothing to the paper under their jointnames. These axioms apparently worked well for themand would probably work well for all collaborators. Inhis obituary of Littlewood, Burkill [1] comments thatusually Littlewood would write the penultimate draftof their joint paper and Hardy would write the oweryprose of which he was a master.

Being at Trinity College in Cambridge, Littlewood en-joyed teaching. His remarks [18] regarding teaching andresearch seem very apropos today. He said, I rmly be-lieve that research should be oset by a certain amountof teaching, if only as a change from the agony of re-search. The trouble ... is that in practice you get eitherno teaching or else far too much.

Of course, Littlewood is not only known for his collab-orative work with Hardy, but his eminent contributionsin other areas of analysis. It seems tting to end thissurvey with a quotation of Littlewood that summarizeshis philosophy: Try a hard problem. You may not solveit, but you will prove something else.


GENERAL ARTICLE

Address for Correspondence

M Ram Murty

Department of Mathematics

Queens University

Kingston, Ontario

K7L 3NR, Canada

Email:

[email protected]

Suggested Reading

[1] J C Burkill, John Edensor Littlewood, Bulletin of the London Math.

Society, Vol.11, pp.59103, 1979.

[2] J E Littlewood, Sur la distribution des nombres premiers, Comptes

Rendus, Vol.158, pp.18691872, 1914.

[3] G H Hardy and J E Littlewood, The zeros of Riemann's zeta function

on the criticial line, Math. Zeit., Vol.10, pp.283317, 1921.

[4] A Selberg, On the zeros of Riemann's zeta function, Skr. Norske Vid.

Akad. Oslo I, Vol.10, p.59, 1942.

[5] N Levinson, More than one-third of the zeros of Riemann's zeta

function are on VVVVV= 1/2, Advances in Math., Vol.13, pp.383436, 1974.[6] B Conrey, More than two-fifths of the zeros of the Riemann zeta

function are on the critical line, J. Reine angew.Math., Vol.399, pp.1

16, 1989.

[7] E C Titchmarsh, On van der Corput's method and the zeta function

of Riemann (V), Quarterly Journal of Math. Oxford, Vol.5, pp.195

210, 1934.

[8] R Balasubramanian, An improvement on a theorem of Titchmarsh on

the mean square of it21] , Proc. London Math. Soc., Vol.36,No.3, pp.540576, 1978.

[9] M Einsiedler, A Katok and E Lindenstrauss, Invariant measures and

the set of exceptions to Littlewood's conjecture, Annals of Math.,

Vol.164, No.2, pp.513560, 2006.

[10] G H Hardy, Ramanujan, Twelve lectures on subjects suggested by his

life and work, Chelsea Publishing Company, New York, 3rd Edition,

1978.

[11] A Selberg, Reflections around the Ramanujan centenary, Resonance,

No.12, pp.8191, 1996. (See also, A Selberg, Collected Papers, Vol.1,

pp.695701, Springer-Verlag, 1989.)

[12] C S Yogananda, Waring's problem and the circle method, Resonance,

Vol.9, No.6, pp.5155, 2004.

[13] S Pillai, Collected Papers, Volume 1, Edited by R Balasubramanian

and R Thangadurai, Ramanujan Mathematical Society, 2010.

[14] K Mahler, On the fractional parts of the powers of a rational number

II, Mathematika, Vol.4, pp.122124, 1957.

[15] R C Vaughan, The Hardy-Littlewood method, 2nd Edition, Cambridge

University Press, 1997.

[16] G H Hardy and J E Littlewood, Some problems of partitio numerorum,

III: On the expression of a number as a sum of primes, Acta Math.,

Vol.44, pp.170, 1922.

[17] D Hensley and I Richards, Primes in intervals, Acta Math., Vol.25,

pp.375391, 1974.

[18] J E Littlewood, Littlewood's Miscellany, Edited by B Bollobs, Cam-

bridge University Press, 1986.

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