SRINIVASA RAMANUJAN ( 1887-1920 ) AND THE THEORY OF PARTITIONS OF NUMBERS AND STATISTICAL MECHANICS A CENTENNIAL

(Received June 5, 1987) ABSTRACT. This centennial tribute commemorates Ramanujan the Mathematician and Ramanujan the Man. A brief account of his llfe, career, and remarkable mathematical contributions is given to describe the gifted talent of Srinivasa RamanuJan. As an example of his creativity in mathematics, some of his work on the theory of partition of numbers has been presented with its application to statistical mechanics.

Finally, Hardy decided to bring RamanuJan to Cambridge in order to pursue some Joint research on mathematics.RamanuJan was pleased to receive an invitation from Hardy to work with him at Cambridge.
But the lack of his mother's permission combined with his strong Hindu religion prejudices forced him to decline Hardy's offer.
As a result of his further correspondence with Hardy, RamanuJan's talent was brought to the attention of the University of Madras.The University made a prompt decision to grant a special scholarship to RamanuJan for a period of two years.On May I, 1913, the 25 year old RamanuJan formerly resigned from the Madras Port Trust Office and Jolned the University of Madras as a research scholar with a small scholarship.He remained in that position until his departure for Cambridge on March 17, 1914.During the years 1903-1914, Ramanuj an devoted himself almost entirely to mathematical research and recorded his results in his own notebooks.Before his arrival in Cambridge, RamanuJan had five research papers to his name, all of which appeared in the Journal of the Indian Mathematical Society.He discovered and/or rediscovered a large number of most elegant and beautiful fornmlas.Thes results were concerned with Bernoulli's and Euler's numbers, hypergeometric series, functional equation for the Riemann zeta function, definite integrals, continued fractions and distribution of primes.
During his stay in Cambridge from 1914 to 1919, RamanuJan worked ontinually together with Hardy and Littlewood on many problems and results conjectured by himself.His close association with two great mathematicians enabled him not only to learn mathematics with rigorous proofs but also to create new mathematics.
Ramanujan was never disappointed or intimidated even when some of his results, proofs or conjectures were erroneous or even false.Absolutely no doubt, he simply enjoyed mathematics and deeply loved mathematical formulas and theorems.It was in Cambridge where his genius burst into full flower and he attained great eminence as a gifted mathematician of the world.Of his thirty-two papers, seven were written in collaboration with Hardy.
Most of these papers on various subjects took shape during the super-productive period of 1914-1919.
These subjects include the theory of partitions of numbers, the Rogers-Ramanujan identities, hyper-geometrlc functions, continued fractions, theory of representation of numbers as sums of squares, Ramanujan's Y-function, elliptic functions and q-serles.
In May 1917, Hardy wrote a letter to the University of Madras informing that Ramanujan was infected with an incurable disease, possibly tuberculosis.In order to get a better medical treatment, it was necessary for him to stay in England for some time more.
In spite of his illness, Ramanujan continued his mathematical research even when he was in bed.
It was not until fall of 1918 that Ramanujan showed any definite sign of improvement.
On February 28, 1918, he was elected a Fellow of the Royal Society at the early age of thirty.He was the first Indian on whom the highest honor was conferred at the first proposal.Niel Bohr was the only other eminent scientist so elected as the Fellow of the Royal Society.On October 13, 1918, he was also elected a Fellow of the Trinity College, Cambridge University with a fellowhslp of 250 a year for the next six years.In his announcement of his election with the award, Hardy forwarded a letter to the Registrar of Madras University by saying, "He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is."He also asked the University to make a permanent arrangement for him in a way which could leave him free for research.The University of Madras promptly responded to Hardy's request by granting an award of 250 a year for five years from April I, 1919 without any duties or assignments.
In addition, the University also agreed to pay all of his travel expenses from England to India.
In the meantime, RamanuJan's health showed some signs of improvement.
So it was decided to send him back home as it deemed safe for him to travel.
Accordingly, he left England on February 27, 1919 and then arrived at Bombay on March 17, 1919.His return home was a very pleasant news for his family, but everybody was very concerned to see his mental and physical conditions as his body had become thin and emaciated.Everyone hoped that his return to his homeland, to his wife and parents and to his friends may have some positive impact on his recovery from illness.Despite his loss of weight and energy, RamanuJan continued his mathematical research even when he was in bed.
In spite of his health gradually deteriorating, RamanuJan spent about nine months in different places including his home town of Kumbakonam, Madras and a village of Kodumudi on the bank of the river Kaveri.
The best medical care and treatment availalbe at that time were arranged for him.Unfortunately, everything was nsuccesful.He died on April 29, 1920 at the age of 32 at Chetput, a suburb of Madras, surrounded by his wife, parents, brothers, friends and admirers.

L. DEBNATH
In his last letter to Hardy on January 12, 1920, three months before his death, RamanuJan wrote: "I discovered very interesting functions recently I call 'Mock' O-functlons.Unlike 'False'0-functlons (studied by Professor Rogers in his interesting paper) they enter into mathematics as beautifully as the ordlnaryO-functions.I am sending you with this letter some examples."Like his first letter of January 1913, Ramanujan's last letter was also loaded with many interesting ideas and results concerning q-serles, elliptic and modular functions.
In order to pay tribute to Srlnlvasa Ramanujan, G.N. Watson selected the contents of RamanuJan's last letter to Hardy along with Ramanujan's five pages of notes on the Mock Theta functions for his 1935 presidential address to the London Mathematical Society.
In his presidential address entitled "The Final Problem: An Account of the Mock Theta Functions", Watson (1936) discussed Ramanujan's results and his own subseqent work with some detail.His concluding remarks included: "Ramanujan's discovery of the Mock Theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end.
As much as his earlier work, the mock theta functions are an achievement sufficient to cause his name to be held in lasting remembrance.
To his students such discoveries will be a source of delight Clearly, RamanuJan's contributions to elliptic and modular functions had also served as the basis of the subsequent developments of these areas in the twentieth century.

RAMANUJAN-HARDY'S THEORY OF PARTITIONS
As an example of RamanuJ an' s creativity and outstanding contribution to mathematics, we briefly describe some of his work on the theory of partitions of numbers and its subsequent applications to statistical mechanics.Indeed, the theory of partitions is one of the monumental examples of success of the Hardy-RamanuJan partnership.
Ramanujan shared his interest with Hardy in the unrestricted partition function or simply the partition function p(n).
This is a function of a positive integer n which is a representation of n as a sum of strictly positive integers.
Thus the map n p(n) defines the partition function.
The number 6 has only one partition into distinct odd parts: 5+I.We also note that there are 4 partitions of 6 into utmost 2 integers, and there are four partitions of slx into integers which do not exceed 2. And there are 3 partitions of six into 2 integers and there are equally 3 partitions of 6 into integers with 2 as the largest.
It follows from the above examples that the value of the partition function p(n) depends on both the size and nature of parts of n.These examples also lead to the concept of restricted and unrestricted partitions of an integer.The restrictions may sometimes be so stringent that some numbers have no partitions at all.For example, I0 cannot be partitioned into three distinct odd parts.
There is a simple geometric representation of partitions which is usually shown by using a display of lattice points (dots) called a Ferret graph.For example, the partition of 20 given by 7+4+4+3+I+I can be represented by 20 dots arranged in five rows as follows: Reading this graph vertically, we get another partition of 20 which is 6+4+4+3+I+I+I.
Two such partitions are called conjugate.
Observe that the part in either of these partitions is equal to the number of parts in the other.This leads to a simple but interesting theorem which states that the number of partitions of n into m parts is equal to the number of partitions of n into parts with m as the largest part.
Several theorems can be proved by simple combinatorial arguments involving graphs.
Above examples with the geometrical representation indicate that partitions have inherent symmetry.In quantum mechanics, such geometric representations of partitions are known as Youn Tableaux which was introduced by Young for his study of symmetric groups.They were also found to have an important role in the analysis of the symmetries of many-electron systems.
The above discussions also illustrate some important and useful role of the partition function from mathematical, geometrical and physical points of view.
In additive questions of the above kind it is appropriate to consider a power series generating function of p(n) defined by From this elementary idea of generating function, Euler formulated the analytical theory of partitions by proving a simple but a remarkable result: where p(o) If 0 < x < and an integer m > and m -I Euler's result (2.2) gives a generating function for the unrestricted partition of an integer n without any restriction on the number of parts or their properties such as size, parity, etc. Hence the generating function for the partition of n into parts with various restriction on the nature of the parts can be found without any difficulty.
For example, the generating function for the partition of n into distinct (unequal) integral parts is Obviously, the right hand side is the generating function for the partition of n into odd integral parts.
Thus it follows from (2.7) and (2.8) that the number of partitions of n into unequal parts is equal to the number of its partitions into odd parts.This is indeed a remarkable result.
Another beautiful result follows from Euler's theorem and has the form The powers of x are the familiar triangular numbers, Ann(n+l) that can be represented geometrically as the number of equidistant points in triangles of different sizes.These points form a triangular lattice.As a generalization of this idea, the square numbers are defined by the number of points in square lattices of increasing size, that is, I, 4,9,16,25 We where Pe(n) is the partition of n into an even number of distinct parts, and Po(n) is the partition of an odd number of distinct parts, and the integers m(n) (3n 2n) are called the Euler pentagonal numbers which can be represented geometrcally by the number of equidistant points in a pentagon of increasing size.
These points form a pentagonal lattice.Also, it follows from (2.2) and (2.10) by actual computation that F(x) (2.11) In view of the fact that the generating functions for the partition of numbers involve infinite products which play an important role in the theory of elliptic and associated functions (Dutta and Debnath, 1965).
Ramanujan made some significant contributions to the theory of partitions.He was not only the first but the only mathematicians who successfully proved several remarkable congruence properties of p(n).All these results are included in his famous conjecture: If p--5,7 or II and 24n---0 (mod pa), > I, then p(n) 0 (rood pa) (2.15)This was a very astonishing conjecture and has led to a good deal of theoretical research and numerical computation on congruence of p(n) using H. Gupta's table (1980) of values of p(n) for n < 300.However, S. Chowla found that this conjecture is not true for n--243.For this n, 24n RamanuJan's conjecture for powers of 7. Finally, A.O.L. Atkin (1967) settled the problem by proving the conjecture for powers of II.RamanuJan's conjecture can now be stated as an important theorem: If 24n-I 0 (mod dr5 a 7 b llC), then pCn) 0 (rood d) (2.17) In connection with his famous discovery of several congruence properties, RamanuJan also studied two remarkable partition identities: (2.20) The functions on the right side of (2.18) (2.19) have power series expansions with integer coefficients, Ramanujan's congruences (2.12) (2.13) follow from these identities.
Subsequently, these identities have created a tremendous interest among many researchers including H.B.C. Darling, L.J. Mordell, H. Rademacher and H.S. Zuckermann.
They proved RamanuJan' s identities by using the theory of modular functions.
Proofs without modular functions were given by D. KruyswlJk (1950) and later by O. Kolberg.The method of Kolberg gave not only the RamanuJan identities but many new ones.
Actual computation reveals that the partition function p(n) grows very rapidly with n.
D.H. Lehmer (1936) computed p(n) for n 14,031 to verify a conjecture of Ramanujan which asserts that p(14,031) 0 (mod 114).This assertion was found to be correct.
This leads to the question of asymptotic representation of p(n) for large n.
,During the early part of the 20th century, Hardy and RamanuJan made significant progress in the determination of an asymptotic formula for p(n).Using elementary arguments, they first showed log p(n) _ ,--, + 0 (n) as n (2.21) Then, with the aid of a Tauberlan Theorem, Hardy and RamanuJan (1918)  RamanuJan and Hardy's second proof was based upon the Cauchy integral formula in complex analysis.In order to outline the proof, we replace real x by a complex z in (2.2) to obtain the Taylor series representation of F(z) and hence the coefficient p(n) of the resulting series can be expressed by the Cauchy integral formula p(n) where F(z) is analytic inside the unit disk Izl in the complex z-plane, C is a suitable closed contour enclosing the origin and lying entirely within the unit disk.It turns out that the unit clrcle is a natural boundary for F(z).
We choose C as a circle with the origin as center and radius r (0 < r < I) in the complex z-plane.We make a change of variable z= exp (2iT) in (2.2) so that  We now rewrite (2.24) in terms of n(T) in the form where F is the line segment of length unity, parallel to the real axis, from 2 -+ ie to + ie, e > 0 and n We also assume C to be such that n 24 F is its image under the transformation z exp(2wiT).
The function n(T) has a simple pole at T=0. RamanuJan and Hardy proved rigorously that the main contribution to the integral for p(n) given by (2.26) comes from the polar singularity at T--0 as n .Thus the asymptotic value of p(n) is given by p where K When is replaced by n, then (2.27) becomes identical with (2.22).
n Finally, H. Rademacher (1937) further improved and fully completed the evaluation of the integral for p(n) by proving an exact formula.He noted that q(T) has also singular at every point T , (p,q) on a segment of the real axis of length unity.He then evaluated contributions to integral (2.26) at its all singular points of the form T p and obtained the exact formula q p(n) ---/--I Z q A (n where A (n) exp(-2ip/q), exp[is(p,q)], (p,q) I, (2.29abc) q p--q P, q P, q and s(p,q) is the Dedekind sum.
The work of RamanuJan-Hardy's partition function combined with that of Rademacher can be regarded as truely remarkable and have stimulated tremendous interests in subsequent developments in the theory of modular functions.
The RamanuJan-Hardy collaboration on the asymptotic analysis for p(n) is one of the monumental results in the history of mathematics and is perhaps best described by J.E. Littlewood (1929) in his review of the collected papers of Srlnlvasa RamanuJan in the Nature.
L. DEBNATH 3. APPLICATIONS TO STATISTICAL MECHANICS One of the most remarkable applications of the RamanuJan-Hardy asymptotic formula for p(n) deals with the problems of statistical mechanics.Several authors including Auluck and Kotharl (1946), Temperley (1949) and Dutta (1956) discussed the significant role of partition functions in statistical mechanics.The theory of partitions of numbers have been found to be very useful for the study of the Bose-Einsteln condensation of a perfect gas.The central problem is the determination of number of ways a given amount of energy can be shared out among different possible states of a thermodynamic assembly.This problem is essentially the same as that of finding the number of partitions of a number into integers under certain restrictions.
We consider a thermodynamic assembly of N non-lnteractlng identical linear simple harmonic oscillators.
The energy levels associated with an oscillator are where n denotes (in units of 00) the energy of the assembly, excluding the residual energy given by the second term of the right side of (3.1).
We denote ?(E) for the number of distinct wave functions assigned to the assembly for the energy state E. It is well known that for a Bose-Einsteln assembly the number of assigned wave functions is the number of ways of distributing n energy quanta among N identical oscillators without any restriction as to the number of quanta assigned to the oscillator.
For a Fermi-Dirac assembly, the energy quanta assigned to all oscillators are all different.
For the case of a classical Maxwell- Boltzmann assembly, oscillators are considered as distinguishable from each other, and the number of wave functions is simply the number of ways of distributing n energy quanta among N distinguishable oscillators.This is equal to the number of ways of assigning N elements to n positions, repetitions of any element are permissible.
If Pd(n) denotes the number of partitions of n into exactly d or less than d parts, then Pd(n)--p(n) for d >_ n where p(n) is the number of partitions of n as a s of positive integers.On the other hand, qd(n) representSdthe number of partitions of n into exactly d unequal parts so that Pd(n) qk(n).On the other k-I hand, the number of partitions of n into exactly d or less different parts is denoted by qd(n) and Qd(n) sands for the number of partitions of n into exactly d unequal parts so that qd(n) [ Qk(n) We also observe the following results: Pd(n) Qd(n + d(d+l)), Qd(n+d) Qd(n) + Qd_t(n), where N! in (3.6) is inserted to make the entropy expression meaningful.
It is important to point out that if N=0(n ), both PN(n) and QN(n+N) tend to N H /N!.This means that for N < < n both the Bose-Einstein statistics and the ermi-Dirac statistics tend to the classical Maxwell-Boltzmann statistics.
The state function Z for an assembly is defined by Z . -, and k is the Boltzmann constant.We can rewrite (3.8)This means that the classical statistics is the limit of both results (3.9) and (3.10).
The above expressions for the state function Z were used to obtain the result for the energy E (or n) and the entropy S.
Using the expression for S, asymptotic formulas for the partition functions PN(n) and p(n) as follows: L. DEBNATH PN (n) 4n3--I exp [7 22 I n exp(-N/6/n)] for N > > n (3.14)Thus, in the limit, n PN(n) p(n) exp (7  (3.15)This is the RamanuJan-Hardy asymptotic formula.
A more general result can be derived in the form PN(n) It is important to point out that this general result for PN(n) reduces to (3.13) as xffiN 0, and to (3.14) as xfN On the other hand, Temperley (1949) applied the RamanuJan-Hardy theory of partitions to discuss results of the Bose-Einsteln condensation theory.He considered a problem different from that of Auluck and Kothari.His model consists of N particles obeying Bose-Einstein statistics distributed among infinitely many energy levels 0, e, 2e, 3, of uniform spacing E in such a way that the total energy is cE.
The partition function PN(E) representing the number of ways of dividing an integer E (energy) into N or less integral parts has the generating function 3 Temperley used the RamanuJan-Hardy asymptotic method to compute PN(E) for large N.
His analysis gives pN(Z) i/ Eexp [7 exp I- This result is similar to that of Auluck and Kotharl (1946) who considered the problem of distribution of a fixed amount of energy between N harmonic oscillators of equal frequency, and solved it from the statistical mechanics of harmonic oscillators.
Temperley also investigated the Bose-Einsteln perfect gas model which is also equivalent to a certain problem in the partition of numbers into sums of squares.The energy levels available to particles in a cubical box of side-length d are given by the expresion where K(r,s,t) (r2+ s2+ t2).Each of these levels may be occupied an integral number of times or not at all.
The problem is to find the asymptotic form for the number of distinct partition of an integer E, representing the ratio of the total to the lowest possible energy-separatlon h2/Smd2, into a sum of the numbers where the order in which the K's are arranged is neglected, but on the other hand, K's like K(1,2,3), K(2,1,3) are distinct from one another, and such K's have to be treated as different, even though they are numerically equal.In all cases, the quantitics r,s,t are positive integers.The upshot of this analysis is the existence of an intermediate temperature region within which the results of earlier theory are unreliable.
It is also confirmed the existence of the phenomena of condensation into lowest energy-levels.At the same time, the present investigation gives a transition departure far below K for a perfect gas of heillum atoms.However, the earlier theory can provide physically sensible results at very high and at very low temperatures.
All these above discussions show a clear evidence for the great importance as well as success of the RamanuJan-Hardy theory of partitions in statistical mechanics.
In an essentially statistical approach to thermodynamic problems, Durra (1953,  1956) obtained some general results from which different statistics viz., those of Bose, Fermi and Gentile, Maxwell-Boltzmann can be derived by using different partitions of numbers.It is noted that mathematical problems of statisitics of Bose, Fermi, and Gentile are those of partitions of numbers (energy) into partitions in which repetition of parts are restricted differently.In partitions corresponding to Bose statistics any part can be repeated any number of times, that to Gentile statistics any part can be repeated upto d times where d is a fixed positive integers, and that to Fermi statistics no part is allowed to repeat, that is, d=l.
All these led to an investigation of a new and different type of partitions of numbers in which repetition of any part is restricted suitably.Motivated by the need of such partition functions and its physical applications to statistical physics, Dutta (1956,   1957)  where p(n) is the unrestricted partition function due to Hardy and RamanuJan (1918).Subsequently, Dutta and Debnath (1957) In particular, for partition of unequal parts (d=l), (3.26)These results are in excellent agreement with those of Hardy and RamanuJan (1918) upto the expotentlal order.The asymptotic result for the unrestricted partition is found to be very useful for computing the dominant term in the expression for the entropy of the corresponding thermodynamic system.
Dutta's partition function is not only more general than that introduced by earlier authors, but also it is more useful for the study of problems in statistical physics.Mathematical problems of Gentile statistics deals with the partitions of numbers (energy) into parts in which repetition of parts are restricted differently.
In partitions corresponding to Bose statistics, any part can be repeated any number of times (d / (R)).The Fermi statistics deals with the partitions of number into parts in which no part can repeat (d=l).In partitions corresponding to Gentile statistics, any part can be repeated up to d times.
In other words, Dutta's partition function d p(n) is found to be useful for an investigation of thermodynamic problems.
As a final example of physical application of RamanuJan-Hardy's theory of partitions of numbers, mention may be made of a paper by Bohn and Kalckar (1937) dealing with calculation of the density of energy levels for a heavy nucleus.

CONCLUDING REMARKS
It is hoped that enough has been discussed to give some definite impression of Ramanujan's great character as well as of the range and depth of his contributions to pure mathematics.Throughout his life, Ramanujan was deeply committed to his family and friends.He also expressed an unlimited interest in education and deep compassion for poor students and orphans who needed support for their education.He also profoundly believed in the dignity and work of human being.
RamanuJan's entire life was totally dedicated to the pursuit of mathematical truth and dissemination of new mathematical knowledge.
His genius was recognized quite early in his llfe and has never been in question.
Indeed, Hardy in his "A Mathematician's Apology" wrote: "I have found it easy to work with others, and have collaborated on a large scale with two exceptional mathematicians (Ramanujan and Littlewood) and this has enabled me to add to mathematics a good deal more than I could reasonably have expected."Also, he said: "All of my best work since then (1911 and 1913) has been bound up with theirs (Ramanujan and Littlewood), ...".There is no doubt at all about RamanuJan's profound and ever-lasting impact on mathematics and mathematical community of the world.
Today, one hundred years after his birth, we pay tribute to this great man, and at the same time, we can assess and marvel at the magnitude of his outstanding achievements.
By any appraisal, RamanuJan was indeed a noble man and a great mathematician of all time.

ACKNOWLEDGEMENTS
The biographical sketch of Srinivasa RamanuJan described in this article is based upon G.H. Hardy's two books entitled Collected Papers of Srinivasa Ramanuan (1927) and Ramanuan (1940).This work was partially supported by the University of Central Florida.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: xm) which is the reciprocal of the generating function of the unrestricted m=l partition function p(n) given by(2.2).This product has the representation (l-xm)
a non-negatlve integer, h=2 is the Planck constant and is the angular frequency.If E denotes the energy of the assembly, a number n ) + QN_I(n) QN(n+N) for the Fermi-Dirac assembly, -N!(N-I)!n! for the Maxwell-Boltzmann assembly,

9 )
For the Fermi-Dirac assembly, we have ZeN Z [QN (n) + QN-I(n) exp(-n)    n:N exp [-N(N-I) ,] (l-er) -I r:l For the classical Maxwell-Boltzmann case, we have -N Ze " N= N I _ .! I-e -It is interesting to point out that as N 2 N-I PN(n N I _ _ !N H exp(2N) n n 27 N 2N for N < < n (3.13) Durra and Debnath (1959) studied a new partition of number n into any number of parts, in which no part is repeated more than d times.Dutta's partition function is denoted by dP(n).Dutta himself and in collaboration with De bnath proved algebraic and congruence properties of dP(n).The generating function of this partition function is 2n xdn) f(x) --I dP(n) xn* (I+ xn+ x + + n--I d+ n(d+l) n l-x n" E (l-x g p(n) x [[ n n(d+l) l-x E (l-x n) r. p(n) x (3.22)

First
Round of Reviews May 1, 2009 (1936)became deeply involved in the proof of the conjecture and also in the computation of p(n) for large n.
Lehmer G.N. Watson (1938) proved This is one of the most remarkable results in the theory of numbers.Equally remarkable was Hardy and RamanuJan's proofs of(2.22).One proof is based on the ) and so ultimately from Euler's table.Using a Tauberlan Theorem, Dutta proved an asymptotic forumla correct up to the exponential order above for d pn) for large n: