Principle of Riemann Zeta Functions and Their Strategies
Exploring Zeta Functions and Their Applications in Number Theory and Algebraic Geometry
by Manjunath C.*,
- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659
Volume 4, Issue No. 7, Nov 2012, Pages 0 - 0 (0)
Published by: Ignited Minds Journals
ABSTRACT
Zeta or L - functions are modelled on the Riemann'szeta function originally defined by the seriesand then extended to the whole complexplane. The zeta function has an "Euler product", a "functionalequation" and though very much studied still keeps secret many of itsproperties, the greatest mystery being the so-called Hiemann Hypothesis. Manysimilar (or thought to be similar) serieshave been introduced in arithmetic,algebraic geometry and even topology, dynamics (we won't discuss the latter).We plan basically to discuss zeta functions attached to algebraic varietiesover finite fields and global fields. The first applications of zeta functions haw been thearithmetic progession theorem (Dirichlet. 1837) "there exists one (hence infinitely) prime congruent to a modulo b.whenever a and b are coprime" and the prime number theorem(Kiemann 1859, with an incomplete proof: Hadamard and de la Vallee Poussin,1896) "the number of primes lessthan x is asymptotic to Butfurther applications were not restricted to the study of prime numbers, theyinclude the study of the ring of algebraic integers, class field theory, theestimation of the size of solutions of (some) diophantine equations, etc.Moreoverhave provided or suggested fundamentallinks between algebraic varieties (motives over Q). Galois representations,modular or au- tomorphic forms: for example, though they do not appearexplicitly in Wiles work, it seems fair to say they played an important role inthe theory that finally led to the solution of Shimura-Taniyama-Weil conjectureand thus of Fermat's Last Theorem. The first four lectures develop results and definitionswhich though all classical are perhaps not too often gathered together. Thefirst lecture introduces Riemann's zeta function, Dirichletassociated to a character. Dedekind zetafunctions and describes some applications of zeta functions: the secondintroduces the Hasse-Weil zeta functions associated to algebraic varietiesdefined over a finite field, a number field or a function field as well as L-functions associated to Galois representations and modular forms; the thirdreviews techniques from complex analysis and estimates for zeta functions: thefourth touches the theory of special values of zeta functions, some known likethe class number formula and some conjectured like the Birch and Swinnerton-Dyer formula. The fifth and final lecture is an exposition of recent work ofBoris Kunyavskif, Micha Tsfasman. Alexei Zykin. Amflcar Facheco and the authoraround versions and analogues of the Brauer-Siegel theorem. Frerequisite will be kept minimal whenever possible : acourse in complex variable and algebraic number t heory, a bit of Galois theoryplus some exposure to algebraic geometry should suffice.
KEYWORD
zeta functions, Riemann's zeta function, Euler product, functional equation, Hiemann Hypothesis, algebraic varieties, finite fields, global fields, arithmetic progression theorem, prime number theorem, ring of algebraic integers, class field theory, diophantine equations, algebraic geometry, Galois representations, modular forms, Shimura-Taniyama-Weil conjecture, Fermat's Last Theorem, Hasse-Weil zeta functions, L-functions, complex analysis, class number formula, Birch and Swinnerton-Dyer formula, Brauer-Siegel theorem
INTRODUCTION
Letbe a finite field of q elements and p its characteristic. Let .V be an algebraic set defined over. For each positive integer k. letdenote the number ofpoints on X. The zeta function Z(X) of X is the generating function The zeta function contains important arithmetic and geometric information concerning X. It has been studied extensively in connection with the celebrated Weil conjectures. Both practical applications and theoretical investigations make a good understanding of the zeta function from an algorithmic point of view increasingly important. The aim of this paper is to present a brief introductory- account of the various fundamental problems and results in the emerging algorithmic theory of zeta functions. We shall focus on general properties rather than on results that are restricted to although in that case one can often say more. The contents are organized as follows. In Section 2 we review general properties of zeta functions from an algorithmic point of view. A naive effective algorithm for computing the zeta function is given. If the characteristic p is small, one can use Dwork'smethod to obtain a polynomial time algorithm for computing the zeta function in the case that the numbers of variables and defining equations for X are fixed. We show that the general case of algebraic sets can be reduced in various ways to the case that A" is a hyper surface with emphasis on the smooth projective case. We consider the complex pure weight decomposition. Using the LLL factorization algorithm and Deligne's main theorem, we show that, when the zeta function is given, one can compute in polynomial time how many zeros and poles with a given complex absolute value it has which is devoted to the P-adic pure slope decomposition, we use the theory of Newton polygons to obtain a similar result for the number of zeros and poles with a givenabsolute value. An algorithm for the simpler problem of computing the zeta function modulo p. This algorithm shares several characteristic features with the general P-adic method for computing the full zeta function that is presented in [Lauder and Wan 2008] in this volume. Section 7 may thus serve as an introduction to that article. All algorithms in this paper are deterministic. Probabilistic algorithms will not be discussed. Time is measured in bit operations.
THE BEHAVIOUR OF
Starting from the formula a reordering of the summations gives, for The last summation writes in the form
(1)
andits fractional part. Notice that the formula (1) is an alternative way to obtain the analytic continuation ofin the half plane When -s = 1, the last integral in (1) is equal to
whereIs the Euler constant.
Finally, formula (1) yields the following asymptotic expansion
(2)
This expansion yields interesting results if one computes the expansion obtained by previous equation By comparison with (2), we obtainIn other words; we have obtained the classical result and the relationyields the beautiful series
GENERALIZED EULER CONSTANTS
The expansion (2) can he continued by writing The constantscan be proved to satisfy These formula generalize the Euler constant definition (corresponding to the case n = 0) and for that reason,
the constantsare often called the generalized
Manjunath C.
informations about these constants can be found on Eric Weissten's world of mathematics site.
STIELTJES AND HADAMARD
From the Euler product formula we have for. It follows that Sincehas a pole at 1.has a root at 1. It turns out that the series actually converges for(van Mangoldt 1897. de la Vallee Poussin 1899). Thus we obtain Euler's curious formula (1748) Let M be the step function defined by M(0) = 0. M piecewise constant, M has a jumpat n. and M Is equal to the average of its left and right limits at each jump. Then If M grows less rapidly thanfor somethen the second integral above will converge absolutely forThen, by uniqueness of analytic contin- uation. We can conclude thathas no roots in. In 1912 Littlewood proved the converse and therefore:
Theorem 1. The Riemann hypothesis is true if and only if for eachwe have
In 1885 Stieltjes wrote to Hermite that he had proved thatis bounded for largeand that therefore the Riemann hypothesis is true. Stieltjes, however, was unable to recall his proof in later years. Hadamard in 1896 published a paper on the zeta only because Stieltjes' proof that there are no zeros inhas not yet been published. It now seems likely that Stieltjes had in fact made an error.
OODS AND ENDS OF EULER RELATION
If we form the Dirichlet product we have where d(n) is the number of divisors of n. We have similar series involving sums of divisors of n or the number of positive integers less than n relatively prime ton.
CONCLUSION
When studying the distribution of prime numbers Riemann extended Euler's zeta function (defined just for s with real part greater than one) to the entire complex plane (sans simple pole at s = 1). Riemann noted that his zeta function had trivial zeros at -2, -4, -6, ...; that all nontrivial zeros were symmetric about the line Re(s) = 1/2; and that the few he calculated were on that line. The Riemann hypothesis is that all nontrivial zeros are on this line. Proving the Riemann Hypothesis would allow us to greatly sharpen many number theoretical results. For example, in 1901 von Koch showed that the Riemann hypothesis is equivalent to:
REFERENCES
[1] A. M. Odlyzko. The 10 to the power 22 zero of the riemann zeta function. In M. van Frankenhuysen and M. L. Lapidus, editors, Dynamical. Spectral, and Arithmetic Zeta Functions, number 290 in Contemporary Math, series, pages 139-144. Amer. Math. Soc., 2001. [3] Edwards, H. M., Riemann's Zeta Function. Academic Press. Inc., San Diego. New York, 1974. [4] Titchmarsh. E. C., The Theory of the Riemann Theta Function, Oxford Univ. Press, London. New York. 1951. [5] Conway, J. B.. Functions of One Complex Variable. Springer-Verlag. New York. Berlin. 1973. [6] Abramowitz, M. and Stegun, I. A. (Editors): Handbook of Mathematical Function* with Formula.*, Graph.s, and Mathematical Tables, Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., 19G4. [7] Ayoub, R.: Euler and the Zeta functions, Arner. Math. Monthly 81, 10G7-108G (1974). [8] Bailey, D.H., Bonvein, J.M., and C'randall, R.E.: On the Khintchine constant, Math. Comput. G6, 417-131 (1997).
