Analytic number theory (nonfiction)

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).

Branches of analytic number theory

Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique.

• Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.
• Additive number theory is concerned with the additive structure of the integers, such as [Goldbach's conjecture (nonfiction)|Goldbach's conjecture]] that every even number greater than 2 is the sum of two primes. One of the main results in additive number theory is the solution to Waring's problem.

History

Precursors

Much of analytic number theory was inspired by the prime number theorem ....

Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function ....

Carl Friedrich Gauss considered the same question: "Im Jahr 1792 oder 1793" ('in the year 1792 or 1793'), according to his own recollection nearly sixty years later in a letter to Encke (1849), he wrote in his logarithm table (he was then 15 or 16) the short note "Primzahlen unter ..." ('prime numbers under ...') .... But Gauss never published this conjecture.

In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li(x) (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(x) and x / ln(x) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.

Dirichlet

Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837 he published Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions. In 1841 he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers ....

Chebyshev

In two papers from 1848 and 1850, the Russian mathematician [[ Pafnuty Chebyshev (nonfiction)|Pafnuty L'vovich Chebyshev]] attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) (for real values of the argument "s", as are works of Leonhard Euler, as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one.[9] He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x.[10] Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2.

Riemann

Bernhard Riemann made some famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he investigated the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.

Hadamard and de la Vallée-Poussin

Extending the ideas of Riemann, two proofs of the prime number theorem were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.

Modern times

The biggest technical change after 1950 has been the development of sieve methods,[13] particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory,[14] which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.

Specifically, the breakthroughs by Yitang Zhang, James Maynard, Terence Tao and Ben Green have all used the Goldston–Pintz–Yıldırım method, which they originally used to prove that ...

Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, the circle method of Hardy and Littlewood was conceived as applying to power series near the unit circle in the complex plane; it is now thought of in terms of finite exponential sums (that is, on the unit circle, but with the power series truncated). The needs of Diophantine approximation are for auxiliary functions that are not generating functions—their coefficients are constructed by use of a pigeonhole principle—and involve several complex variables. The fields of Diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture.

Problems and results

Theorems and results within analytic number theory tend not to be exact structural results about the integers, for which algebraic and geometrical tools are more appropriate. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate.

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Methods of analytic number theory

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See also

• [[Automorphic L-function (nonfiction)
• [[Automorphic form (nonfiction)
• [[Langlands program (nonfiction)
• [[Maier's matrix method (nonfiction)

External links

• Analytic number theory @ Wikipedia