For odd square-free n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and square-free, n=2 ( x 2 ) = C 2 n nxD 2 n ; Here A n (x); : : : ; D n (x) are polynomials with integer coecients. We show how these coef- cients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n (x); : : : ; D n (x), and illustrate the application to integer factorization with some numerical examples.

The Wiener--Khintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a Wiener--Khintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PACS: 05.40+j; 02.30.Nw; 02.10.Lh Keywords: Twin primes; Ramanujan--Fourier series; Wiener--Khintchine formula 1. Introduction " The Wiener--Khintchine theorem states a relationship between two important characteristics of a random process: the power spectrum of the process and the correlation function of the process" [1]. One of the outstanding problems in number theory is the problem of prime pairs which asks how primes of the form p and p+h (where h is an even integer) are distributed. One immediately notes that this is a problem of #nding correlation between primes. We make two key observations. First of all there is an arithmetical function (a function de#ned on integers) which traps the...

this paper is to discuss each of the Lambert series identities as well as to provide the arithmetical corollaries to which Ramanujan alluded. Several are related to the number of representations of an integer as a sum of squares or as a sum of triangular numbers. One of the most interesting identities yields a formula for the number of ways an integer can be represented as a sum of six triangular numbers. Although we expect that this formula is in the classical literature, we have been unable to find it there. The only appearances of this formula known to us appear in papers of V. G. Kac and M. Wakimoto [14] in 1994 and K. Ono, S. Robins, and P. T. Wahl [20] in 1995. Let r k (n) denote the number of ways the positive integer n can be represented as a sum of k squares, with representations arising from different signs and from different orders being regarded as distinct. By convention, r k (0) = 1: Also, let

Ideas from physics are used to show that the prime pairs have the density conjectured by Hardy and Littlewood. The proof involves dealing with infinities like in quantum field theory. Keywords: twin primes, Poisson summation formula, Ramanujan - Fourier expansion, renormalisation 1 E-mail: padma@imsc.ernet.in 2 E-mail: padma@imsc.ernet.in 1 This article may be considered as an invitation to theoretical physicists to enter the field of additive number theory. For sometime now there have been serious attempts to cross fertilise the disciplines of physics and number theory. It seems strange that on the one hand the most practical of disciplines, namely, physics has connections with the most aracane of disciplines, namely, number theory. However, surprising connections have appeared between number theory and physics as can be seen in [1], [3] and [9]. The work of Ramanujan in particular has had surprising connections with string theory, conformal field theory and statistical physics. For sometime now the authors one of whom is a theoretical physicist and the other a number theorist have been trying to understand problems in additive number theory using ideas from both fields. One such problem is the distribution of prime pairs. Prime pairs are numbers which are primes differing by some even integer. For example, 3; 5; 5; 7; 11; 13; 17; 19 and so on are all prime pairs with common difference 2. The question is whether such prime pairs are infinite, if so, what is the density? We will now summarise the standard method used to attack the problem which is the circle method. For technical reasons, the von Mangoldt function (n) (which is defined to be log p if n = p m where p is a prime and 0 otherwise) is used instead of the characteristic function on the primes. Hence in...

Introduction The classical Hermite Reciprocity Law asserts the isomorphism S m S n (k 2 ) = S n S m (k 2 ) of symmetric powers of representations of the Lie group SL 2 (k) acting standardly on k 2 , for a characteristic zero field k (see [6], Remark 12 by V. L. Popov in Appendix 3 of the Russian translation). In particular, the space of degree m polynomial invariants of the irreducible (n+1)-dimensional representation is equidimensional with the space of degree n invariants of the irreducible (m + 1)-dimensional representation. Recently in [3] there was obtained an explicit formula for the dimension a 0 (n; m) of the space of degree m homogeneous pol

For odd square-free n ? 1 the cyclotomic polynomial \Phi n (x) satisfies the identity of Gauss 4\Phi n (x) = A 2 n \Gamma (\Gamma1) (n\Gamma1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is \Phi n ((\Gamma1) (n\Gamma1)=2 x) = C 2 n \Gamma nxD 2 n or, in the case that n is even and square-free, \Sigma\Phi n=2 (\Gammax 2 ) = C 2 n \Gamma nxD 2 n ; Here An (x); : : : ; Dn (x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for An (x); : : : ; Dn (x), and illustrate the application to integer factorization with some numerical examples.

Recall that the Möbius µ-function from elementary number theory is defined so that µpnq p 1q k if n is a product of k distinct primes, and µpnq 0 if n is divisible by the square of a prime. (So µp1q p 1q 0 1.) For any arithmetic function f (i.e., any f: N Ñ C), its Dirichlet transform fp is defined by pfpnq: fpdq, and its Möbius transform q f by qfpnq: d|n µpn{dqfpdq. d|n The well-known Möbius inversion formula ([2, Theorems 266, 267]) says precisely that the qpf pqf. Möbius and Dirichlet transforms are inverses of each other: for any f, we have f Our proof of the infinitude of primes is based on the following lemma. By the support of f, we mean the set of natural numbers n for which fpnq 0. Lemma (Uncertainty principle for the Möbius transform). If f is an arithmetic function which does not vanish identically, then the support of f and the support of q f cannot both be finite. Proof. Suppose for the sake of contradiction that both f and fq are of finite support. Let 8¸ F pzq fpnqz n. Then F is entire (in fact, a polynomial function). On the other hand, for |z| (1) F pzq

Abstract. We prove a simple formula for the main value of r-even functions and give applications of it. Considering the generalized Ramanujan sums cA(n, r) involving regular systems A of divisors we show that it is not possible to develop a Fourier theory with respect to cA(n, r), like in the the usual case of classical Ramanujan sums c(n, r).

Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be Ramanujan- Fourier series. 1 Introduction. The two well known methods in additive number theory are the circle method and the sieve method. The circle method is based on using a generating function (See Section 3) and noting along with Ramanujan and Hardy that the rational points on the circle contribute most and then through estimates showing that the contribution from the other points is

Ramanujan sums analysis of long-period sequences