The 1896 Proof of Prime Number Theorem

The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem

Possibly the single most important problem in Analytic Number Theory, and certainly the driving force behind many of its' results, is the question of the distribution of the prime numbers. The proof, at the end of the nineteenth century, of what is now known as the Prime Number Theorem is one of mathematics' greatest achievements, representing the culmination of the work of some of history's finest mathematicians. Although the Prime Number Theorem has been proved in many different ( but all essentially equivalent ) ways, every analytic proof to date relies on properties of the Riemann zeta-function, illustrating the pivotal role this function plays.

The distribution of the prime numbers, like so many other areas of mathematics, was first investigated by Euclid, who proved that the number of primes is infinite. However, the more immediate origin of Riemann's work came in a series of investigations by Leonhard Euler who proved, in 1737, that the sum of the reciprocals of the primes
i.e. 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ........... is a divergent series, indicating the greater density of distribution of the primes than, say, the squares ( the sum of 1 + 1/4 + 1/9 + 1/16 + ....... being a convergent series ) . Euler also observed that this sum of reciprocals of primes, not exceeding x, seems to be
asymptotically equal to log( log x ) and, most importantly for Riemann's work, he also proved the identity which bears his name:

EULER'S IDENTITY

 inf   1 /                  1 /
SUM    / r   =    PRODUCT   / ( 1 - 1 / r )  
 n=1   / n             p      /           / p

                with r real, p prime

Futher important work was done by Carl Friederich Gauss and Adrien-Marie Legendre in the last decade of the eighteenth century, both mathematicians observing that pi( x) - the number of primes not exceeding x - seems to be
asymptotic to x/( log x ) . Writing this in modern terms:

THE PRIME NUMBER THEOREM

           lim   ( pi(x).log(x)/x )  =  1
        x -> inf

However, neither mathematician could provide a proof of this, the proof also eluding Peter Gustave Lejeune Dirichlet, who had, nevertheless, made many important discoveries in the field.

Pafnuti Liwowich Chebyshev made the next significant contribution when he showed, in 1851, that IF the above limit does exist, then that limit must be 1. Again, though, the proof of the existence of this limit was the problem and, although Chebyshev's work was an immense stride forward, he faced the same insurmountable hurdle that all of his predeccors in the struggle had faced in that he was restricted to working with a step- function, whether it be pi( x ) or his own psi( x ), both constant between consecutive integers.

In 1859 Bernhard Riemann wrote his momentous 8-page memoir entitled
"Ueber die Anzahl der Primzahlen unter einer Gegebenen Grosse"
("On the Number of Primes Less than a Given Magnitude");
the revolution in thinking that this paper introduced was to relate the study of the primes to the properties of a CONTINUOUS complex variable - his zeta function - which, following on from the work of Euler, Riemann defined as:-

RIEMANN'S ZETA FUNCTION

                  1 /                 inf 1 /
ZETA(s) = PRODUCT  / ( 1 - 1/ s) = SUM  / s
                p      /          / p     n=1 / n

               with p prime, s complex

In this memoir, Riemann made some striking discoveries about zeta( s ), many of which were to prove crucial to the proof of the Prime Number Theorem, including:-
(i) Zeta( s ) is single-valued and finite ( i.e. analytic ) for all values of s other than 1.
(ii) Zeta( s ) vanishes when s is a negative even integer.
(iii) ALL other zeros of zeta( s ) lie within a 'critical strip' of values of u between 0 and 1, are not real, and are symmetrically placed about the x-axis and the line s=1/2.
Most fundamentally of all the connection was made between the distribution of the primes and the zeros of zeta( s ) in the critical strip, a direct link between prime integers and complex numbers. This was to prove vital as, in 1896, almost simaltaneously Jacques Hadamard and Charles de le Vallee Poussin obtained proofs of the Prime Number Theorem. For these proofs a zero-free region of
zeta ( s ) : u = 1 - c/( log t ) , with t greater than 2, was used ( de la Vallee Possin having obtained a proof that, for u = 1, zeta( u + it ) does not vanish for real t ), and an explicit relationship for pi( x ) in terms of log ( zeta ( s ) ) was found ( where the zeros of zeta ( s ) are obviously of critical importance ). The integrals so obtained were considered around some closed rectangular contour, estimating the integrals along the bounds of this contour, both of these mathematicians having chosen different contours, thus supplying different proofs of the Prime number Theorem.

All subsequent analytic prrofs of the theorem have depended upon certain properties of Riemann's zeta function. Estimates are made of either log( zeta( s ) ) or quotient of the derivative of zeta( s ) divided by zeta( s ) by considering integrals over contours in the half-plane u greater than 1/2, which can be extended by analytic continuation, but avoiding the point 1.

It should be noted that although 'elementary' proofs of the theorem exist, the first being given by Paul Erdos and Atle Selberg in 1949, these proofs - which are far from elementary in the grammatical sense of the word - do not lead to such good estimates of pi( x ) as do the analytic proofs, so that the analytic proofs retain their central interest.

Another, essential, aspect of Riemann's paper which has yet to be mentioned, but without which no study of the role of Riemann's zeta function in the proof of the Prime Number Theorem would be complete, is the well-known 'Riemann hypothesis'. In his 1859 paper Riemann states that he finds it 'very likely' that all of the complex zeros of zeta( s ) have real part u=1/2, although he writes "i have put aside the search for such a proof after some fleeting vain attempts". Despite a great deal of circumstancial evidence in favour of Riemann's hypothesis ( the first 3,000,000 points at which zeta( s ) = 0 for u between 0 and 1 ALL lie on the line u=1/2 ), no proofs exist, and indeed there is evidence to suggest that other zeros may exist. The accuracy of the estimate for pi( x ) given by the most refined versions of the Prime Number Theorem is limited, but could be greatly improved if the Riemann Hypothesis was proven. Indeed, in 1902 David Hilbert listed this problem amongst the most important problems facing mathematicians at the beginning of the twentieth century. The same comment still applies at the beginning of the twenty-first century.

Overall, the real contribution of Riemann's paper, astonishingly the only one he ever published in the field of Number Theory before his untimely death at the age of 39, lay not in its' results but in it's treatment of the zeta function as a complex variable. The impact of his paper can, perhaps, best be measured that in the next 30 years after publication, little progress was made, almost as if it took that long to fully comprehend what Riemann had indicated. Very few papers in the history of mathematics have left their mark so indelibly upon the subject, and Riemann's zeta function - in particular the Riemann Hypothesis - remains the most-studied field of analytic number theory, 140 years after its' introduction.

BIBLIOGRAPHY

1. T.M.Apostol - An introduction to analytic number theory

2. H.M.Edwards - Riemann's zeta function

3. A.Ivic - The Riemann zeta function

4. E.Landau - Collected works, Vol 1

5. M.B.Nathanson - Elementary methods in number theory

6. S.J.Patterson - An introduction to the theory of the Riemann zeta function

7. H.E.Rose - A course in number theory

8. E.C.Titchmarsh - The theory of the Riemann zeta function

+ Open University MSc. course M829 - supplement on the Prime Number Theory

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The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem

EULER'S IDENTITY

SUM / r = PRODUCT / ( 1 - 1 / r ) n=1 / n p / / p with r real, p prime

n=1 / n p / / p with r real, p prime

Futher important work was done by Carl Friederich Gauss and Adrien-Marie Legendre in the last decade of the eighteenth century, both mathematicians observing that pi( x) - the number of primes not exceeding x - seems to be asymptotic to x/( log x ) . Writing this in modern terms:

THE PRIME NUMBER THEOREM

RIEMANN'S ZETA FUNCTION

BIBLIOGRAPHY

SUM / r = PRODUCT / ( 1 - 1 / r )
n=1 / n p / / p with r real, p prime

Futher important work was done by Carl Friederich Gauss and Adrien-Marie Legendre in the last decade of the eighteenth century, both mathematicians observing that pi( x) - the number of primes not exceeding x - seems to be
asymptotic to x/( log x ) . Writing this in modern terms: