An elementary proof of the primenumber theorem for. Introduction in this paper will be given a new proof of the primenumber theorem, which is elementary in the sense that it uses practically no analysis, except the simplest properties of the logarithm. At the same time atle selberg published his paper, an elementary proof of the primenumber theorem, in the annals of mathematics s. I learned it when i started learning analytic number theory. Selberg and paul erdos each obtained elementary proofs of the prime number theorem, both using the asymptotic formula above as a starting point. Ideas in the elementary proof of the prime number theorem selberg.
We shall prove the primenumber theorem in the form 1. Quotefromweylslettertoselbergaugust31,1948 isitnottruethatyouwereinpossessionofwhaterd. Newman found a theorem with a short proof that provided a much simpler link between the zeta function and the prime number theorem. In this article, we discuss the first elementary proof, due to selberg and erd. An elementary proof of the primenumber theorem for arithmetic progressions volume 2 atle selberg. In the rst section, selberg introduces the prime number theorem as. Selberg and on a new method in elementary number theory which leads to an elementary proof of the prime number theorem, by p. Alte selberg first proved the prime number theorem elementarily. The fact that the prime number theorem may be proven using only elementary methods is something of an intellectual. In 1949 he gave an elementary but by no means simple proof of the prime number theorem, a result that had theretofore required advanced theorems from analysis.
Letting pn denote the number of primes p b n, gauss conjectured in the early. Unfortunately, these proofs are still much longer than the shortest proofs of today that use complex analysis. To this, selberg replied that erdoss discovery was essential in selbergs e orts to create an elementary proof of the prime number theorem. Proof of the prime number theorem joel spencer and ronald graham p rime numbers are the atoms of our mathematical universe. Circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between the two mathematicians. An elementary proof was found by erdos 1949 and selberg 1950 ball and coxeter 1987, p. He tried to talk selberg into providing a seminar, showing the power of his inequality by giving an elementary proof of dirichlets theorem on primes in arithmetic progressions. The prime number theorem, that the number of primes selbergs proof to summarize selbergs proof of the prime number theorem, selberg divides the proof into four di erent sections.
Analysis of selbergs elementary proof of the prime number theorem josue mateo historical introduction prime numbers are a concept that have intrigued mathematicians and scholars alike since the dawn of mathematics. Several different proofs of it were found, including the elementary proofs of atle selberg and paul erdos 1949. An elementary proof of the primenumber theorem, by a. The basic new thing in the proof is a certain assymptotic formula 2. Convergence theorems the rst theorem below has more obvious relevance to dirichlet series, but the second version is what we will use to prove the prime number theorem. Analysis of selbergs elementary proof of the prime number. Denote the mobius function, which is defined as follows. The constants implied by the os are generally dependent on k and in sees. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Versions of elementary proofs of the prime number theorem appear in final. Version 1 suppose that c nis a bounded sequence of. An elementary proof of the prime number theorem 7 for t 2. To summarize selbergs proof of the prime number theorem, selberg divides the proof into four dierent sections.
Many of selbergs papers were published in number theory, trace formulas and discrete groups 1989. An elementary proof of the primenumber theorem lsu math. The two eventually collaborated to create a rudimentary proof of the prime number theorem, which relied solely on an extensive use of logarithms. In 1980, newman gave a new, simple proof of the prime number theorem, based on complex. A prime number is a natural number greater than 1 that has no positive divisors besides 1 and itself. On selbergs elementary proof of the primenumber theorem. The complexanalytic proof of the prime number theorem can help inform the elementary one. From this formula there are several ways to deduce the primenumber theorem. By july of that year, selberg and paul erdos had each obtained elementary proofs of the pnt, both using selbergs. Prime number elementary proof tauberian theorem joint paper prime. During his proof, he found the following identity, known as the selberg identity. In 1948, alte selberg and paul erdos simultaneously found elementary proofs of the prime number theorem. Analytic number theory seemed like the core of numbers to me.