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Ito, Y. The largest prime factor of integers in the short interval; Chaohua Jia. A general divisor problem in Landau's framework; S. Kanemitsu, A. Asymptotic expansions of double gamma-functions and related remarks; K.
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Analytic Number Theory Jia Matsumoto 2002
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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 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.
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 ,  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 ,  which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
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.xn----itbjbanp5adgf8b0d.xn--p1ai/scripts/11.php
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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. 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. 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.
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. In particular, they proved that if. This remarkable result is what is now known as the prime number theorem. 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. Also, it has been proven unconditionally i. The general case was proved by Hilbert in , using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by Hardy and Littlewood.
These techniques are known as the circle method, and give explicit upper bounds for the function G k , the smallest number of k th powers needed, such as Vinogradov 's bound. Diophantine problems are concerned with integer solutions to polynomial equations: one may study the distribution of solutions, that is, counting solutions according to some measure of "size" or height.
In geometrical terms, given a circle centered about the origin in the plane with radius r , the problem asks how many integer lattice points lie on or inside the circle. In general, an O r error term would be possible with the unit circle or, more properly, the closed unit disk replaced by the dilates of any bounded planar region with piecewise smooth boundary.
One of the most useful tools in multiplicative number theory are Dirichlet series , which are functions of a complex variable defined by an infinite series of the form. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane.
The utility of functions like this in multiplicative problems can be seen in the formal identity.
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Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series or a product of simpler Dirichlet series using convolution identities , examine this series as a complex function and then convert this analytic information back into information about the original function.
Euler showed that the fundamental theorem of arithmetic implies at least formally the Euler product. Euler was also the first to use analytical arguments for the purpose of studying properties of integers, specifically by constructing generating power series. This was the beginning of analytic number theory.
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There is a plethora of literature on this function and the function is a special case of the more general Dirichlet L-functions. Analytic number theorists are often interested in the error of approximations such as the prime number theorem.