Introduction

Let $\pi(x)$ denote the number of primes not exceeding the real number $x$. In 1793 C.F. Gauß [7] and in 1798 A.M Legendre [19] proposed independently that for large $x$ the ratio

$$\frac{\pi(x)}{x/\log x}$$

was nearly $1$ and they conjectured that this ratio would approach $1$ as $x$ approaches $\infty$. Both Gauß and Legendre attempted to prove this statement but did not succeed. The problem of deciding the truth or falsehood of this conjecture attracted the attention of eminent mathematicians for nearly 100 years.

In 1851 the russian mathematician P.L. Chebychev [1] made an important step forward by proving that if the ratio did tend to a limit, then this limit must be one. Further, he succeeded in showing that the actual order of $\pi(x)$ is $x/\log x$, that is

$$\pi(x)\asymp \frac{x}{\log x}.$$

However, he was unable to prove that the ratio does tend to a limit.

In 1859 B. Riemann [20] attacked the problem with analytic methods, using a formula discovered by L. Euler in 1737 which relates the prime numbers to the function

$$\zeta(s):=\sum_{n=1}^{\infty}\frac{1}{n^s}=\prod_{p\;{\rm prime}}\left( 1-\frac{1}{p^s}\right) ^{-1}$$

for real $s>1$. Riemann considered complex values of $s$ and outlined an ingenious method for connecting the distribution of primes to properties of the function $\zeta(s)$. The mathematics needed to justify all the details of his method had not been fully developed and Riemann was unable to completely settle the problem before his death in 1866.

Thirty years later the necessary analytic tools were at hand and in 1896 J. Hadamard [9] and C.J de la Vallée Poussin [23] independently and almost simultaneously succeeded in proving that

$$\pi(x) \sim \frac{x}{\log x} \hspace{1cm} {\rm as} \hspace{0,2cm} x \to \infty.$$

This remarkable result is called the prime number theorem, and its proof was one of the crowning achievements of analytic number theory.

The prime number theorem was subsequently reproved and improved by others. However, a proof of this theorem, not fundamentally dependent upon the ideas of the theory of functions, seemed, not only to G.H. Hardy (cf. [11] p.549-550), extraordinarily unlikely. In his talk on "Goldbach`s Theorem" given for the Mathematical Society in Kopenhagen on October 6, 1921 he says:

"...Let us turn back ...to its central theorem, the `Primzahlsatz` or `prime number theorem`...No elementary proof is known, and one may ask whether it is reasonable to expect one. Now we know, that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann`s Zeta-function has no zeros on a certain line. A proof of such a theorem, not fundamentally dependent upon the ideas of the theory of function, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way ...If anyone produces an elementary proof of prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed ..."

Therefore, in 1949 A. Selberg [21] and P. Erdös [4] caused a sensation when they discovered an elementary proof of the prime number theorem. Their proof, though very intricate, makes no use of $\zeta(s)$ nor of complex function theory and in principal is accessible to anyone familiar with elementary analysis.

In 1911 E. Landau [18] showed that the prime number theorem is equivalent to the validity of the assertion that the mean value ${\bf M}(\mu)$ of the Möbius function $\mu$ exists and is equal to zero. We say that the function $f$ possesses a mean value ${\bf M}(f)$ if the limit

$${\bf M}(f): = \lim_{x \to\infty}\hspace{0,1cm}\frac{1}{x}\;\sum_{n\leq x} \; f(n)$$

exists. Here the function $\mu$ is multiplicative, i.e. $\mu(mn) = \mu(m) \mu(n)$ whenever $(m,n)(:=gcd(m,n))=1$, and defined by

$$\mu(1) = 1$$

$$\mu(p^{m}) = \left\{\begin{array}{ll} -1 & ,{\rm if} \;m = 1 \\ 0 & ,{\rm if} \; m > 1.\\ \end{array} \right.$$

As mentioned above the (first) elementary proof of the prime number theorem appeared in 1949. In 1943, A. Wintner [24] , in his book on Erathostenian averages, asserted that if a multiplicative function $f$ may have only values $\pm1$, then the mean value ${\bf M}(f)$ always existed. But, the sketch of his proof could not be substantiated, and the problem remained open as the Erdös-Wintner conjecture. We shall not repeat the story concerning the prize which Erdös offered for a solution of this problem (cf. [3] , p. 254) but in 1967, E. Wirsing [25] solved this problem. His proof was done by elementary methods and thus, he gave another elementary proof of the prime number theorem. He showed that every real-valued multiplicative function $f, \vert f\vert\leq1$, possesses a mean value, but he could not handle the complex-valued case in its full generality. Only by an analytic method, found by G. Halász in 1968 [10] the asymptotic behaviour of $$\sum_{n\leq x}\;f(n)$$ could be fully determined for all complex-valued multiplicative functions $f$ of modulus smaller than or equal to one. It took 24 years until H. Daboussi and K.-H. Indlekofer [2] produced an elementary proof of Halász`s theorem (see [17] ).

Proposition (Halász's theorem). Let $f: {\mathbb{N}}\to {\mathbb{C}}$ be multiplicative, $\vert f\vert \leq 1.$ If there exists a real number $a_0$ so that the series

$$\sum_{p}p^{-1} (1-Ref(p)p^{-ia})$$ (1)

converges for $a=a_0$, then, as $x \to \infty$,

$$x^{-1} \sum_{n \leq x} f(n)=\frac{x^{ia_0}}{1+ia_0} \prod_{p \leq x}(1-p^{-1})(1+\sum_{m=1}^{\infty} p^{-m(1+ia_0)}f(p^m)) +o(1).$$ (2)

If the series (1) diverges for all $a \in {\mathbb{R}}$, then

$$x^{-1} \sum_{n \leq x} f(n)=o(1) \hspace{1cm} (x \to \infty).$$

In either case there are constants $c,a_0$ and a slowly oscillating function $L(u)$ with $\vert L(u)\vert=1$, so that, as $x \to \infty,$

$$x^{-1} \sum_{n \leq x} f(n)=cx^{ia_0} L(\log x) +o(1).$$

Bibliography