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19
On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree 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 squarefree, n=2 ( x 2 ) ..."
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Cited by 14 (5 self)
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For odd squarefree 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 squarefree, 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.
RamanujanFourier series, the WienerKhintchine formula and the distribution of prime pairs
, 1999
"... The WienerKhintchine 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 WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PA ..."
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Cited by 4 (2 self)
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The WienerKhintchine 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 WienerKhintchine 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; RamanujanFourier series; WienerKhintchine formula 1. Introduction " The WienerKhintchine 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...
Which Cayley graphs are integral
 Electronic J. Comb
"... Let G be a nontrivial group, S ⊆ G \ {1} and S = S −1: = {s −1  s ∈ S}. The Cayley graph of G denoted by Γ(S: G) is a graph with vertex set G and two vertices a and b are adjacent if ab −1 ∈ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all co ..."
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Cited by 3 (1 self)
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Let G be a nontrivial group, S ⊆ G \ {1} and S = S −1: = {s −1  s ∈ S}. The Cayley graph of G denoted by Γ(S: G) is a graph with vertex set G and two vertices a and b are adjacent if ab −1 ∈ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs. 1
Fragments by Ramanujan on Lambert Series
"... 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 identiti ..."
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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
2008 Convolution and crosscorrelation of RamanujanFourier series Preprint 0805.0284 [math.NT
"... Abstract. One of the remarkable achievements of Ramanujan[1], Hardy[2] and Carmichael[3] was the development of RamanujanFourier series which converge to an arithmetic function. The RamanujanFourier series, for an arithmetic function, a(n), is given by ∞X a(n) = aqcq(n) q=1 where the Ramanujan su ..."
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Cited by 2 (0 self)
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Abstract. One of the remarkable achievements of Ramanujan[1], Hardy[2] and Carmichael[3] was the development of RamanujanFourier series which converge to an arithmetic function. The RamanujanFourier series, for an arithmetic function, a(n), is given by ∞X a(n) = aqcq(n) q=1 where the Ramanujan sum, cq(n), is defined as qX cq(n) = k=1 (k,q)=1 e 2πin k q and (k, q) is the greatest common divisor of k and q. The object of this paper is to show that if two, arithmetic functions, a(n) and b(n), have RamanujanFourier series of, ∞X ∞X a(n) = aqcq(n) and b(n) = bqcq(n) q=1 then a crosscorrelation between a(n) and b(n) can be given based on the RamanujanFourier coefficients. Specifically: 1 (1) lim N→ ∞ N n=1 q=1 NX a(n + m) ¯ ∞X b(n) = aq¯bqcq(m) where ¯x is the complex conjugate of x This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and crosscorrelation. The general results for two almost periodic functions are narrowed and applied to Ramanujan sums and finally applied to support the specific relation (1). The WienerKhinchin fomula, (1), connecting the autocorrelation of an arithmetic function and the coefficients of its Ramanujan Fourier series is a powerful link between the circle method and the sieve methods found in number theory. The application of this WeinerKhinchin formula to number theory is described in the works of H. G. Gadiyar and R. Padma [4], [5], [6], [7]. The WienerKhinchin fomula is used in [4] to prove the HardyLittlewood conjecture and is used in [6] to prove the density of Sophie primes. q=1
Renormalisation and the Density of Prime Pairs
, 1998
"... 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 Email: padma@im ..."
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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 Email: padma@imsc.ernet.in 2 Email: 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...
Combinatorics Of Necklaces And "hermite Reciprocity"
"... 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 ..."
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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
On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree 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 ..."
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For odd squarefree 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 squarefree, \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.
THE MÖBIUS TRANSFORM AND THE INFINITUDE OF PRIMES
"... Recall that the Möbius µfunction 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 ..."
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Recall that the Möbius µfunction 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: dn µpn{dqfpdq. dn The wellknown 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
REMARKS ON GENERALIZED RAMANUJAN SUMS AND EVEN FUNCTIONS
, 2006
"... Abstract. We prove a simple formula for the main value of reven 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 us ..."
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Abstract. We prove a simple formula for the main value of reven 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).