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In 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. Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory, [3] 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 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 two papers from and , the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. Riemann's statement of the Riemann hypothesis, from his paper. 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.

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He made a series of conjectures about properties of the zeta function , one of which is the well-known Riemann hypothesis. The biggest technical change after has been the development of sieve methods , [10] 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 , [11] which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.

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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.

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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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Euclid showed that there are infinitely many prime numbers. An important question is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. Gauss , amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number N is close to the value of the integral.

Dirichlet series for dynamical systems of first-order ordinary differential equations

Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture. Riemann found that the error terms in this expression, and hence the manner in which the primes are distributed, are closely related to the complex zeros of the zeta function.

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In particular, they proved that if. This remarkable result is what is now known as the prime number theorem.

Dirichlet series for dynamical systems of first-order ordinary differential equations

It is a central result in analytic number theory. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression with a and q coprime contains infinitely many primes. The prime number theorem can be generalised to this problem; letting.

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