H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1994, p. 8; MR1297543 (96i:11002).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....have a central role to play in minimization problems in the supremum norm as well as many other extremal problems. See, for example, 6] The analogous problem where the polynomials are restricted to have integer coe#cients is very much harder. For a very nice discussion of this problem see [15]. We define (1.4) n [a, b] and let (1.5) # [a, b] inf # n [a, b] n=0,1, lim n### n [a, b] We call any polynomial p n that achieves# n [a, b]annth integer Chebyshev polynomial on [a, b] The above limit exists and equals the infimum because 1, 2 1 n b as follows ....

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, Vol. 84, Amer. Math. Soc., R. I., 1980.

....(2.4) and thus the estimate j Z (ut)j j Z ( 1 u)t)j du: 2.5) To complete the proof, we give a series of lemmas. Lemma 2.2. For any real numbers y and z, the random variable h y;z (U) de ned by (1.2) satis es jE e ithy;z (U) j 2jtj Proof. This follows by a method of van der Corput [2, 13, 6], using little more than the fact that h y;z is convex with h y;z 8 on (0; 1) Lemma 2.3. For any random variable Z and real t, we have j SZ (t)j 2jtj . Proof. Lemma 2.2 yields j SZ (t)j = ith Z;Z (U) E ith Z;Z (U) Z; Z 2jtj Returning to our sequence (Z n ) ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Reg. Conf. Ser. Math. 84, AMS, Providence, R.I., 1994.

....a proof can be obtained by finding a sequence of weights w n with B(w n ) ##. On the other hand, we did not observe a numerical improvement of the estimate (1. 11) when using further factors of the one dimensional integer Chebyshev polynomials for the weight w, beyond the factors x and 1 x (see [19], 7] and [22] Thus one needs a better insight into the arithmetic nature of such factors, to address the problem stated below. Problem 1.4. For w(x) as in (1.12) and # = i=1 # i m i , find (1.15) B : sup . If B = 1 then find a sequence of weights that gives this value. If B 1 then ....

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, Vol. 84, Amer. Math. Soc., Providence, R.I., 1994.

....set E # R 2 must satisfy # E x # dx = # (2) for all # # 1 # . Thus the result of [7] # = 46 73 #) implies that if E is Steinhaus then (2) holds for all # 46 27 ; this is the best that we know unconditionally. The conjectured result (# = 1 2 #, see e.g. 8] or [11]) on (1) would imply (2) for all # 1. This same range # 1 also arises in another way see the remark after the proof of Corollary 2.3. Property (2) with # = 2 can be proved by an argument similar to [9] but based on L 2 # L 2 instead of L 1 # L # estimates. We give this argument in ....

....space W 1 2 , i.e. the integral # R d # # #E (#) 2 d# must be infinite. In fact, there is an asymptotic expression which implies in particular that # # #R # #E (#) 2 d# # R 1 (15) as R # #. This is often used in connection with irregularities of distribution; see e.g. [11]. We will not use (15) in this paper, but we will need to know that the lower bound in (15) is valid without any regularity assumptions on the set E. This is not di#cult but does not seem to be in the literature, so we prove it in Corollary 2.2 below. Let # be a Schwarz class function in R d ....

H. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society, 1994. 28

....nite eld with p k elements. Let f : F p k C be a nontrivial multiplicative character, and extend f to a function on F p k by letting f(0) 0. It is then easy to see that the following holds: 1.1) X x2F p k f(x)f(x h) 1 if h 6= 0 p k 1 if h = 0 Cohn asked (see p. 202 in [3]) if the converse is true in the following sense: if a function f : F p k C satis es (1.2) f(0) 0; f(1) 1; and jf(x)j = 1 for x 6= 0 and equation 1.1, does it follow that f is a multiplicative character The problem has recently received some attention. In [2] Choi and Siu proved that the ....

H. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, American Mathematical Society, Providence, RI, 1994. Department of Mathematics, University of Georgia, Athens GA 30602 (kurlberg@math.uga.edu)

....j SZ (t)j Z 1 0 j Z (ut)j j Z ( 1 u)t)j du: 2.5) To complete the proof, we give a series of lemmas. Lemma 2.2. For any real numbers y and z, the random variable h y;z (U) de ned by (1.2) satis es jE e ithy;z (U) j 2jtj 1=2 : Proof. This follows by a method of van der Corput [2, 13, 6], using little more than the fact that h y;z is convex with h 00 y;z 8 on (0; 1) Lemma 2.3. For any random variable Z and real t, we have j SZ (t)j 2jtj 1=2 . Proof. Lemma 2.2 yields j SZ (t)j = E e ith Z;Z (U) E E e ith Z;Z (U) Z; Z ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Reg. Conf. Ser. Math. 84, AMS, Providence, R.I., 1994.

....that equality holds in (1.17) but this remains open (essentially the same conjecture was also made in [8, p. 90] One may try to construct various sequences of polynomials F n # F n , n # N, to obtain lower bounds for t Z ( 0, 1] from (1. 17) A few of such sequences have been devised (cf. [22] and [8] with the best known being the Gorshkov sequence of polynomials. It was originally found by Gorshkov in [17] and rediscovered by Wirsing [22] and others. These polynomials arise as the numerators in the sequence of iterates of the rational function u(x) x(1 x) 1 3x(1 x) ....

....of polynomials F n # F n , n # N, to obtain lower bounds for t Z ( 0, 1] from (1.17) A few of such sequences have been devised (cf. 22] and [8] with the best known being the Gorshkov sequence of polynomials. It was originally found by Gorshkov in [17] and rediscovered by Wirsing [22] and others. These polynomials arise as the numerators in the sequence of iterates of the rational function u(x) x(1 x) 1 3x(1 x) and they give the following lower bound: 1.18) t Z ( 0, 1] # 1 s 0 = 0.420726 . see [22, pp. 183 188] The upper bounds for t Z ( 0, ....

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, vol. 84, Amer. Math. Soc., Providence, R.I., 1994.

....is sup y D(m; 0; y) These two are equivalent up to multiplicative constants: 1 2 D(m) sup y jD(m; 0; y)j D(m) The sum in Question 6.1 may be expressed as P M k=1 f (k ) where f (x) fx g 1 2 . Since f is of total variation 2 on the torus [0; 1) Koksma s inequality ([4] p.1 3) allows us to bound the sum in terms of the discrepancy of the sequence fk g. Theorem 7.2 (Koksma s Inequality) Let f be a function of bounded variation on the torus [0; 1) Then: m X j=1 f(u j ) m Z 1 0 f(t)dt 1 2 D(m) V ar(f) Proof. Since dD(m; 0; ....

H. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CMBS Regional Conference Series in Mathematics no. 84, AMS, Providence, Rhode Island, 1994.

....that equality holds in (1.17) but this remains open (essentially the same conjecture was also made in [8, p. 90] One may try to construct various sequences of polynomials Fn 2 Fn ; n 2 N; to obtain lower bounds for t Z ( 0; 1] from (1. 17) A few of such sequences have been devised (cf. [22] and [8] with the best known being the Gorshkov sequence of polynomials. It was originally found by Gorshkov in [17] and rediscovered by Wirsing [22] and others. These polynomials arise as the numerators in the sequence of iterates of the rational function u(x) x(1 x) 1 3x(1 x) 6 IGOR ....

....sequences of polynomials Fn 2 Fn ; n 2 N; to obtain lower bounds for t Z ( 0; 1] from (1.17) A few of such sequences have been devised (cf. 22] and [8] with the best known being the Gorshkov sequence of polynomials. It was originally found by Gorshkov in [17] and rediscovered by Wirsing [22] and others. These polynomials arise as the numerators in the sequence of iterates of the rational function u(x) x(1 x) 1 3x(1 x) 6 IGOR E. PRITSKER and they give the following lower bound: 1.18) t Z ( 0; 1] 1=s 0 = 0:420726 : see [22, pp. 183 188] The upper bounds for t Z ....

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, vol. 84, Amer. Math. Soc., Providence, R.I., 1994.

....(t) # # 1 0 # Z (ut) # Z ( 1 u)t) du. 2.5) To complete the proof, we give a series of lemmas. Lemma 2.2. For any real numbers y and z, the random variable h y,z (U) defined by (1.2) satisfies E e ithy,z (U) # 2 t 1 2 . Proof. This follows by a method of van der Corput [2, 13, 6], using little more than the fact that h y,z is convex with h ## y,z # 8 on (0, 1) Lemma 2.3. For any random variable Z and real t, we have # SZ (t) # 2 t 1 2 . Proof. Lemma 2.2 yields # SZ (t) # # #E e ith Z,Z #(U) # # # # E # # #E # e ith Z,Z # (U) # # # Z, Z # ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Reg. Conf. Ser. Math. 84, AMS, Providence, R.I., 1994.

.... an interval I of T and an integer d 0, the Selberg majorizing polynomial is a trigonometric polynomial of degree at most d which majorizes the characteristic function of the interval I, and which is a good approximation to the characteristic function of I in the L 1 norm (see Chapter 1 of [M]) Specifically there is a trigonometric polynomial S d (x; I) with Char I (x) S d (x; I) such that S d (0) Z T S d (x; I)dx = jIj 1 d 1 : Thus taking each f j = S d j (x; I j ) in Proposition 3.1 we deduce the following result. 14 ANDREW GRANVILLE AND K. ....

H.L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, C.B.M.S. Regional Conference Ser. Math, vol. 84, Amer. Math. Soc, 1994.

....bounds are not sharp (except in the trivial case of c 0 ) they are the best that we can get without too much work, but we expect that substantial improvements are possible. Proof. The basic approach is to use the fundamental relation (1. 1) We will first show, using a method of van der Corput [1, 10], that the characteristic function of h y,z (U) is bounded by 2 t 1 2 for each y, z. Mixing, this yields Theorem 2.1 for p = 1 2. Then we will use another consequence of (1.1) namely, the functional equation #(t) Z 1 u=0 #(ut) #( 1 u)t) e itg(u) du, t # R, 2.5) or rather its ....

....equation #(t) Z 1 u=0 #(ut) #( 1 u)t) e itg(u) du, t # R, 2.5) or rather its consequence #(t) # Z 1 u=0 #(ut) #( 1 u)t) du, 2.6) and obtain successive improvements in the exponent p. We give the details as a series of lemmas, beginning with a standard calculus estimate [10]. Note that it su#ces to consider t 0 in the proofs because #( t) #(t) and thus #( t) #(t) Note also that the best constants satisfy c p = sup t 0 t p #(t) although we do not know in advance of proving Theorem 2.1 that these are finite) and thus c 1 p p = sup t 0 t #(t) ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Reg. Conf. Ser. Math. 84, AMS, Providence, R.I., 1994.

....estimate jOE S(Z) t)j Z 1 u=0 jOE Z (ut)j jOE Z ( 1 Gamma u)t)j du: 2.5) We give a series of lemmas. Lemma 2.2. For any real numbers y and z, the random variable h y;z (U) defined by (1.1) satisfies jEe ithy;z (U) j 2jtj Gamma1=2 : Proof. This follows by a method of van der Corput [1, 10, 6], using only the fact that h y;z is convex with h 00 y;z 8 on (0; 1) Lemma 2.3. For any random variable Z and real t, jOE S(Z) t)j 2jtj Gamma1=2 . Proof. Lemma 2.2 yields jOE S(Z) t)j = fi fi fiEe ith Z;Z (U) fi fi fi E fi fi fiE i e ith Z;Z (U) fi fi fi Z; Z jfi fi fi ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Reg. Conf. Ser. Math. 84, AMS, Providence, R.I., 1994.

....bounds are not sharp (except in the trivial case of c 0 ) they are the best that we can get without too much work, but we expect that substantial improvements are possible. Proof. The basic approach is to use the fundamental relation (1. 1) We will first show, using a method of van der Corput [1, 10], that the characteristic function of h y;z (U) is bounded by 2jtj Gamma1=2 for each y; z. Mixing, this yields Theorem 2.1 for p = 1=2. Then we will use another consequence of (1.1) namely, the functional equation OE(t) Z 1 u=0 OE(ut) OE( 1 Gamma u)t) e itg(u) du; t 2 R; 2.5) or rather ....

....OE(t) Z 1 u=0 OE(ut) OE( 1 Gamma u)t) e itg(u) du; t 2 R; 2.5) or rather its consequence jOE(t)j Z 1 u=0 jOE(ut)j jOE( 1 Gamma u)t)j du; 2.6) and obtain successive improvements in the exponent p. We give the details as a series of lemmas, beginning with a standard calculus estimate [10]. Note that it suffices to consider t 0 in the proofs because OE( Gammat) OE(t) and thus jOE( Gammat)j = jOE(t)j. Note also that the best constants satisfy c p = sup t 0 t p jOE(t)j (although we do not know in advance of proving Theorem 2.1 that these are finite) and thus c 1=p p = sup ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. CBMS Reg. Conf. Ser. Math. 84, AMS, Providence, R.I., 1994.

....by the assumption, often made in cryptography, that the triples (g x , g y , g xy ) cannot be distinguished from totally random triples in feasible computation time. See Section 2 for more details. We note that the uniformity referred to above (which is for example described in Chapter 1 of [21]) is not quite the same as the notion of uniform probability distribution more familiar to workers in cryptography. Another interpretation of (the quantitative form of) our result is the relative independence of any portion # 1 48 of the most significant bits of the smallest non negative ....

H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Math. Vol. 84, Amer. Math. Soc., Providence, 1994.

....O(B) We also denote by log u the binary logarithm of a real u and put e p (a) exp(2#ia p) a # IF p . Thus e p (a) is a non trivial additive character of IF p . We need a form of the Erdos Turan inequality which relates the discrepancy and character sums, see Corollary 1. 1 to Chapter 1 of [10] or Corollary 3.11 of [13] Lemma 1 For any set M # IF p the bound D(M) # 1 p 1 #M p 1 X h=1 1 h X m#M e p (hm) holds. 4 We also need the following upper bound on character sums with exponential functions which is essentially Theorem 3.4 of [7] Lemma 2 Let p be prime ....

H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis , CBMS Regional Conference Series in Math. Vol. 84, Amer. Math. Soc., Providence, 1994.

....a similar result when x( n ) hence the amount of rotation at each time step, is changing slowly. The machinery to quantify this idea comes from the theory of uniform distribution. The first inequality we use is the following combination of Koksma s inequality and Erd os Tur an inequality [7]: For any finite sequence (u n ) n2I in [0; 1] X n2I un 1 2 jI j 1 2 min K 0 jI j K 1 3 K X k=1 1 k X n2I e 2 ikun : 25) We apply this inequality to the finite sequence wN0 1 ; wN0 n . The next step is to bound the exponential sums ....

....That is, wm = m) m = N 0 ; N 0 1; N 0 n: 27) We construct such a in [6] see also Proposition 1 below. Once wm is rewritten in this form, we replace the exponential sum by something that is easier to estimate. We use the following truncated Poisson summation formula from [7], which is originally due to van der Corput: Theorem 3 (Truncated Poisson) Let f be a real valued function and suppose that f 0 is continuous and increasing on [a; b] Put = f 0 (a) f 0 (b) Then X a m b e 2 if(m) X 1 1 Z b a e 2 i(f( d O(log(2 ) ....

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H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, AMS, 1994.

....O(B) We also denote by log u the binary logarithm of a real u and put e p (a) exp(2 ia=p) a 2 IF p : Thus e p (a) is a non trivial additive character of IF p . We need a form of the Erdos Tur an inequality which relates the discrepancy and character sums, see Corollary 1. 1 to Chapter 1 of [8] or Corollary 3.11 of [11] Lemma 1. For any set M IF p the bound D(M) 1 p 1 #M p Gamma1 X h=1 1 h fi fi fi fi fi X m2M e p (hm) fi fi fi fi fi holds. We also need the following upper bound on character sums with exponential functions which is essentially Theorem 3.4 of [5] ....

H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Math. Vol. 84, Amer. Math. Soc., Providence, 1994.

....for up to 10 7 points in dimensions 6. By some modifications of Roth s proof, the bound (4) can be improved somewhat, but so far I haven t succeeded in getting considerably better values. Few other techniques exist for lower bounding the L 2 discrepancy and its variants (Beck [1] Montgomery [14]) but so far they do not look very promising in this respect either. 2 An explicit bound calculated by Kuipers and Niederreiter [12] pp. 103 104) from their version of Roth s proof, which is 2 Gamma8d (log 2 n= d Gamma 1) d Gamma1 , is considerably smaller and it seems to be ....

H. L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis (CBMS Regional conference series in mathematics No. 84). Amer. Math. Soc., Providence, 1994.

....ae R 2 must satisfy Z E jxj ff dx = 1 (2) for all ff fi 1 Gammafi . Thus the result of [7] fi = 46 73 ffl) implies that if E is Steinhaus then (2) holds for all ff 46 27 ; this is the best that we know unconditionally. The conjectured result (fi = 1 2 ffl, see e.g. 8] or [11]) on (1) would imply (2) for all ff 1. This same range ff 1 also arises in another way see the remark after the proof of Corollary 2.3. Property (2) with ff = 2 can be proved by an argument similar to [9] but based on L 2 L 2 instead of L 1 L 1 estimates. We give this argument in ....

....space W 1 2 , i.e. the integral R R n j j j c E ( j 2 d must be infinite. In fact, there is an asymptotic expression which implies in particular that Z j jR j c E ( j 2 d R Gamma1 (15) as R 1. This is often used in connection with irregularities of distribution; see e.g. [11]. We will not use (15) in this paper, but we will need to know that the lower bound in (15) is valid without any regularity assumptions on the set E. This is not difficult but does not seem to be in the literature, so we prove it in Corollary 2.2 below. Let OE be a Schwarz class function in R d ....

H. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society, 1994.

....(C k ) k2N converges to a limit C; and Borwein and Erd elyi [3] showed that C 2 (0:8586616; 0:8657719) Therefore one cannot prove the prime number theorem in this way. However the problem of finding the integer Chebyshev polynomials in the interval [0; 1] is interesting in itself (See [3, 5] and the references therein. In particular, Borwein and Erd elyi state in [3] that Even computing low degree examples is complicated. In this paper, we first prove two lemmas that halve the degree of the polynomials we need to look for. This step enables us to compute polynomials of larger ....

Montgomery, H. L. Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, vol. 84 of CBMS Regional Conference Series in Mathematics. AMS, Oct. 1994.

....was found in [8] fl m = i sec 2(m 1) j m 1 , and it is interesting to note that a properly scaled Chebyshev polynomial of the first kind is extremal. For our purpose, a simplified version, namely, the estimate m 1 2 m is sufficient. The latter was proved by H.L. Motgomery (see [7], Ch. 5) Let us take P (z) Q N j=1 p(zz Gamma1 j ) where p(z) 2 P (1;0) m is the mth H al asz extremal polynomial. Then obviously P (z) 2 P 1 mN ; P (0) 1; P (z j ) 0; j = 1; N ; max jzj1 jP (z)j (m ) N i 1 2 m j N e 2N m , which proves (46) A polynomial ....

H.L. Montgomery. Ten lectures on the interface between analytic number theory and harmonic analysis. expository lectures from the NSF-CBMS Regional Conference held at Kansas State University, Manhattan, Kansas, May 22-25, 1990. (Regional conference series in mathematics, ISSN 0160-7642; no.84), AMS, 220 pp.

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H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1994, p. 8; MR1297543 (96i:11002).

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H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conf. Ser. in Math. 84, Amer. Math. Soc., Providence, RI, 1994.

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H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS, vol. 84, Amer. Math. Soc., Providence, R.I., 1994. 14 IGOR E. PRITSKER

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H.L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS No. 84, 1994.

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H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Math. Vol. 84, Amer. Math. Soc., Providence, 1994.

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H.L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, vol. 84, CBMS Regional Conference Series in Mathematics, AMS, 1994.

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H. L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS, Regional Conference Series in Mathematics, Number 84, American Mathematical Society, Providence, Rhode Island, 1994.

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