Inmathematics,analytic number theory is a branch ofnumber theory that uses methods frommathematical analysis to solve problems about theintegers.[1] It is often said to have begun withPeter Gustav Lejeune Dirichlet's 1837 introduction ofDirichletL-functions to give the first proof ofDirichlet's theorem on arithmetic progressions.[1][2] It is well known for its results onprime numbers (involving thePrime Number Theorem andRiemann zeta function) andadditive number theory (such as theGoldbach conjecture andWaring's problem).
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.[3]
Much of analytic number theory was inspired by theprime number theorem. Let π(x) be theprime-counting function that gives the number of primes less than or equal tox, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states thatx / ln(x) is a good approximation to π(x), in the sense that thelimit of thequotient of the two functions π(x) andx / ln(x) asx approaches infinity is 1:
known as the asymptotic law of distribution of prime numbers.
Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the functiona/(A ln(a) + B), whereA andB are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture, withA = 1 andB ≈ −1.08366.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 1838Peter Gustav Lejeune Dirichlet came up with his own approximating function, thelogarithmic 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) andx / ln(x) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory,[6] 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 publishedDirichlet's theorem on arithmetic progressions, usingmathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced theDirichlet characters andL-functions.[6][7] In 1841 he generalized his arithmetic progressions theorem from integers to thering ofGaussian integers.[8]
In two papers from 1848 and 1850, the Russian mathematicianPafnuty 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 ofLeonhard 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)) asx 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 allx.[10] Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to proveBertrand's postulate that there exists a prime number betweenn and 2n for any integern ≥ 2.
"…es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien."
"…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation."
Bernhard Riemann made some famous contributions to modern analytic number theory. Ina single short paper (the only one he published on the subject of number theory), he investigated theRiemann zeta function and established its importance for understanding the distribution ofprime numbers. He made a series of conjectures about properties of thezeta function, one of which is the well-knownRiemann hypothesis.
Extending the ideas of Riemann, two proofs of theprime number theorem were obtained independently byJacques Hadamard andCharles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods fromcomplex 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 variables that have the forms = 1 + it witht > 0.[12]
The biggest technical change after 1950 has been the development ofsieve methods,[13] particularly in multiplicative problems. These arecombinatorial 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 isprobabilistic 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 byYitang Zhang,James Maynard,Terence Tao andBen Green have all used theGoldston–Pintz–Yıldırım method, which they originally used to prove that[15][16][17][18][19][20]
Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, thecircle method ofHardy andLittlewood was conceived as applying topower series near theunit circle in thecomplex 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 ofDiophantine approximation are forauxiliary functions that are notgenerating functions—their coefficients are constructed by use of apigeonhole principle—and involveseveral complex variables. The fields of Diophantine approximation andtranscendence theory have expanded, to the point that the techniques have been applied to theMordell 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 numberN is close to the value of theintegral
In 1859Bernhard Riemann used complex analysis and a specialmeromorphic function now known as theRiemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x. 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. Using Riemann's ideas and by getting more information on the zeros of the zeta function,Jacques Hadamard andCharles Jean de la Vallée-Poussin managed to complete the proof of Gauss's conjecture. In particular, they proved that ifthen
This remarkable result is what is now known as theprime number theorem. It is a central result in analytic number theory. Loosely speaking, it states that given a large numberN, the number of primes less than or equal toN is aboutN/log(N).
More generally, the same question can be asked about the number of primes in anyarithmetic progressiona +nq for any integern. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression witha andq coprime contains infinitely many primes. The prime number theorem can be generalised to this problem; lettingthen ifa andq are coprime,where is thetotient function.[21]
There are also many deep and wide-ranging conjectures in number theory whose proofs seem too difficult for current techniques, such as thetwin prime conjecture which asks whether there are infinitely many primesp such thatp + 2 is prime. On the assumption of theElliott–Halberstam conjecture it has been proven recently that there are infinitely many primesp such thatp + k is prime for some positive evenk at most 12. Also, it has been proven unconditionally (i.e. not depending on unproven conjectures) that there are infinitely many primesp such thatp + k is prime for some positive evenk at most 246.
One of the most important problems in additive number theory isWaring's problem, which asks whether it is possible, for anyk ≥ 2, to write any positive integer as the sum of a bounded number ofkth powers,
The case for squares,k = 2, wasanswered by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares. The general case was proved byHilbert in 1909, using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem byHardy andLittlewood. These techniques are known as the circle method, and give explicit upper bounds for the functionG(k), the smallest number ofkth powers needed, such asVinogradov'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" orheight.
An important example is theGauss circle problem, which asks for integers points (x y) which satisfy
In geometrical terms, given a circle centered about the origin in the plane with radiusr, the problem asks how manyinteger lattice points lie on or inside the circle. It is not hard to prove that the answer is, where as. Again, the difficult part and a great achievement of analytic number theory is obtaining specific upper bounds on the error term E(r).
It was shown by Gauss that. In general, anO(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. Furthermore, replacing the unit circle by the unit square, the error term for the general problem can be as large as a linear function of r. Therefore, getting anerror bound of the formfor some in the case of the circle is a significant improvement. The first to attain this wasSierpiński in 1906, who showed. In 1915, Hardy andLandau each showed that one doesnot have. Since then the goal has been to show that for each fixed there exists a real number such that.
In 2000Huxley showed[22] that, which is the best published result.
One of the most useful tools in multiplicative number theory areDirichlet series, which are functions of a complex variable defined by an infinite series of the form
Depending on the choice of coefficients, this series may converge everywhere, nowhere, or on some half plane. In many cases, even where the series does not converge everywhere, theholomorphic 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
hence the coefficients of the product of two Dirichlet series are themultiplicative convolutions of the original coefficients. Furthermore, techniques such aspartial summation andTauberian 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 thefundamental theorem of arithmetic implies (at least formally) theEuler product
where the product is taken over all prime numbersp.
Euler's proof of the infinity ofprime numbers makes use of the divergence of the term at the left hand side fors = 1 (the so-calledharmonic series), a purely analytic result. Euler was also the first to use analytical arguments for the purpose of studying properties of integers, specifically by constructinggenerating power series. This was the beginning of analytic number theory.[20]
Later, Riemann considered this function for complex values ofs and showed that this function can be extended to ameromorphic function on the entire plane with a simplepole ats = 1. This function is now known as the Riemann Zeta function and is denoted byζ(s). There is a plethora of literature on this function and the function is a special case of the more generalDirichlet L-functions.
Analytic number theorists are often interested in the error of approximations such as the prime number theorem. In this case, the error is smaller thanx/log x. Riemann's formula for π(x) shows that the error term in this approximation can be expressed in terms of the zeros of the zeta function. Inhis 1859 paper, Riemann conjectured that all the "non-trivial" zeros of ζ lie on the line but never provided a proof of this statement. This famous and long-standing conjecture is known as theRiemann Hypothesis and has many deep implications in number theory; in fact, many important theorems have been proved under the assumption that the hypothesis is true. For example, under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is.
In the early 20th centuryG. H. Hardy andLittlewood proved many results about the zeta function in an attempt to prove the Riemann Hypothesis. In fact, in 1914,Hardy proved that there were infinitely many zeros of the zeta function on the critical line
This led to several theorems describing the density of the zeros on the critical line.
On specialized aspects the following books have become especially well-known:
Certain topics have not yet reached book form in any depth. Some examples are(i)Montgomery's pair correlation conjecture and the work that initiated from it,(ii) the new results of Goldston, Pintz and Yilidrim onsmall gaps between primes, and(iii) theGreen–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist.
| International | |
|---|---|
| National | |
