## The Gelfond–Schnirelman method in prime number theory (2005)

Venue: | Canad. J. Math |

Citations: | 5 - 5 self |

### BibTeX

@ARTICLE{Pritsker05thegelfond–schnirelman,

author = {Igor E. Pritsker},

title = {The Gelfond–Schnirelman method in prime number theory},

journal = {Canad. J. Math},

year = {2005},

pages = {1080--1101}

}

### OpenURL

### Abstract

Abstract. The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshev-type lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev’s ψ-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1 Lower Bounds for Arithmetic Functions Let π(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that (1.1) π(x) ∼ x log x as x → ∞. We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see, e.g., [8, 10, 17, 29]). Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x log x

### Citations

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Citation Context ...op the ideas of [21] and [7], and establish a connection with the weighted potential theory (or potential theory with external fields) that originated in the work of Gauss [14] and Frostman [13] (see =-=[25]-=- for a modern account on this theory). An important part of the method is the analysis of the asymptotic behavior for the supremum norms of the weighted Vandermonde determinants (1.10), which is gover... |

170 |
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Citation Context ...Let S := �L l=1 [al, bl]. For any TL−1 ∈ RL−1[x], we have � 1 TL−1(t) dt (3.8) πi (t − z) � R(t) = � 0, z ∈ S, TL−1(z)/ � R(z), z ∈ C \ S. S Proof. A detailed proof of this =-=known fact may be found in [20] -=-(cf. Chapter 11). We give a sketch of argument based on Cauchy integral formula. Consider a contour Γ that consists of L simple closed curves around each of the intervals [al, bl], located close to t... |

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Citation Context ...p([a, b], w) := e −Vw . In agreement with the notation of Section 1, we set cw := cap([0, 1], w). If w ≡ 1 on [a, b], then we obtain the classical logarithmic capacity cap([a, b], 1) = (b − a)/4=-= (cf. [23]).-=- The support Sw plays a crucial role in determining the equilibrium measure µw itself, as well as other components of this weighted energy problem. Indeed, if Sw is known then µw can be found as a s... |

134 |
die Anzahl der Primzahlen unter einer gegebenen
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Citation Context ..., [17], [8], [29] and [10]). Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x x ≤ π(x) ≤ 1.106 as x → ∞. log x log x The famous Riemann=-=’s paper [24],-=- published in 1859, gave a strong impulse to the study of complex analytic methods related to the zeta function. Thus Hadamard and de la Vallée Poussin independently proved the Prime Number Theorem i... |

133 | Multiplicative Number Theory - Davenport - 1980 |

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Citation Context ..., 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 (se=-=e [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 α = � k i=1 αimi, find (1.15) B... |

68 |
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Citation Context ...type weights was suggested by Chudnovsky [7] and developed further by Amoroso [1]. For more general weights, one should consult Chapter IV of [25] and the paper of Deift, Kreicherbauer and McLaughlin =-=[9]. We-=- give an explicit form of the equilibrium measure and describe its support in the following result.s6 IGOR E. PRITSKER Theorem 2.1. Let Z := �K i=1 {zi} ⊂ [a, b] be the set of zeros for w of (2.1)... |

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Citation Context ...1) . = cap([a, b], w) (see Theorem III.1.3 of [25]). The quantity on the left-hand side of (2.17) is called the weighted transfinite diameter of [a, b]. In the case w ≡ 1, it was introduced by Feket=-=e [12] -=-for arbitrary compact sets in the plane. Szegő [28] showed that the transfinite diameter coincides with the logarithmic capacity, so that (2.17) is a generalization of his result. We quantify the rat... |

64 | General Orthogonal Polynomials
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Citation Context ... L� [al, bl], l=1 Tzj (x) dx πi(x − zj) � , x ∈ R(x) L� [al, bl], l=1 �sTHE GELFOND-SCHNIRELMAN METHOD IN PRIME NUMBER THEORY 11 Proof. These formulas are essentially known (see, e.g., Le=-=mma 4.4.1 of [27] and Lemma-=- 2.3 of [30]). It is possible to deduce (3.9)-(3.10) from Lemma 3.1, which is done below. tions: We select T∞(t) = � L−1 j=1 cjt j ∈ RL−1[t] so that it satisfies the following equa(3.11) and... |

61 |
Computational Excursions in Analysis and Number Theory, Springer–Verlag 2002
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Citation Context ...omery [19, Ch. 10] (also see Chudnovsky [7]). We are led by this method to the so-called integer Chebyshev problem on polynomials with integer coefficients minimizing the sup norm (see, e.g., Borwein =-=[5]). Let Zn[x] b-=-e the set of polynomials over integers, of degree at most n. In view of (1.6)-(1.7), we are interested in the integer Chebyshev constant (1.8) tZ([0, 1]) := lim n→∞ � inf 0�≡pn∈Zn[x] �pn... |

53 |
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Citation Context ...re and Gauss, states that (1.1) π(x) ∼ x as x → ∞. log x We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see,=-= e.g., [17], [8], [29]-=- and [10]). Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x x ≤ π(x) ≤ 1.106 as x → ∞. log x log x The famous Riemann’s paper [24], ... |

25 | Multiplicative Number Theory. Second edition - DAVENPORT - 1980 |

22 |
Pluripotential theory, London Mathematical Society Monographs
- Klimek
- 1991
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Citation Context ...ssed. Among the natural candidates are the multivariate Vandermonde determinants (see [3, 32]) and other sequences of minimal polynomials [4]. This subject is closely related to pluripotential theory =-=[18]-=-. 2 Potential Theory With External Fields We consider a special case of the weighted energy problem on a segment of the real line [a, b] which is associated with the “polynomial-type” weights (1.12). ... |

16 |
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Citation Context ...(x) dx πi(x − zj) � , x ∈ R(x) L� [al, bl], l=1 �sTHE GELFOND-SCHNIRELMAN METHOD IN PRIME NUMBER THEORY 11 Proof. These formulas are essentially known (see, e.g., Lemma 4.4.1 of [27] and Le=-=mma 2.3 of [30]). It is possi-=-ble to deduce (3.9)-(3.10) from Lemma 3.1, which is done below. tions: We select T∞(t) = � L−1 j=1 cjt j ∈ RL−1[t] so that it satisfies the following equa(3.11) and (3.12) � al+1 bl L−1 ... |

15 |
Elementary methods in the study of the distribution of prime numbers
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- 1982
(Show Context)
Citation Context ... that (1.1) π(x) ∼ x as x → ∞. log x We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see, e.g., [17], [8], [2=-=9] and [10]). Chebyshe-=-v [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x x ≤ π(x) ≤ 1.106 as x → ∞. log x log x The famous Riemann’s paper [24], published in 1859, g... |

14 |
Bemerkungen zu einer Arbeit von Herrn M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten
- Szegö
- 1924
(Show Context)
Citation Context ... The quantity on the left-hand side of (2.17) is called the weighted transfinite diameter of [a, b]. In the case w ≡ 1, it was introduced by Fekete [12] for arbitrary compact sets in the plane. Szeg=-=ő [28]-=- showed that the transfinite diameter coincides with the logarithmic capacity, so that (2.17) is a generalization of his result. We quantify the rate of convergence in (2.17).s8 IGOR E. PRITSKER Lemma... |

13 |
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- Tenenbaum, France
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Citation Context ...s, states that (1.1) π(x) ∼ x as x → ∞. log x We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see, e.g., [17]=-=, [8], [29] and [10]).-=- Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x x ≤ π(x) ≤ 1.106 as x → ∞. log x log x The famous Riemann’s paper [24], published i... |

11 |
On a New Method in Elementary Number Theory Which Leads to an Elementary proof of the Prime Number Theorem
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Citation Context ...does not have zeros on the line {1 + it, t ∈ R}. But the “elementary” approaches to the PNT, which do not use complex analysis and the zeta function, still remained attractive. Selberg [26] and =-=Erdős [11]-=- found the first elementary proof of the Prime Number Theorem in 1949. A survey of elementary methods, with detailed history, may be found in Diamond [10]. The subject of this paper is the elementary ... |

11 | An Elementary proof of the Prime Number Theorem - Selberg - 1949 |

11 |
Transfinite diameter, Chebyshev constants, and capacity for compacta in
- Zaharjuta
- 1975
(Show Context)
Citation Context ...to design other sequences of polynomials providing good bounds for the arithmetic functions, along the discussed lines. Among the natural candidates are the multivariate Vandermonde determinants (see =-=[32]-=- and [3]) and other sequences of minimal polynomials [4]. This subject is closely related to pluripotential theory [18]. 2. Potential theory with external fields We consider a special case of the weig... |

10 |
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(Show Context)
Citation Context ...so that 1/n (1.7) lim �pn� n→∞ [0,1] ? = 1/e, then the PNT followed from (1.6). A nice account on the original Gelfond-Schnirelman attempt is contained in Montgomery [19, Ch. 10] (also see Chu=-=dnovsky [7]-=-). We are led by this method to the so-called integer Chebyshev problem on polynomials with integer coefficients minimizing the sup norm (see, e.g., Borwein [5]). Let Zn[x] be the set of polynomials o... |

8 |
f -transfinite diameter and number theoretic applications,” Ann
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Citation Context ...E GELFOND-SCHNIRELMAN METHOD IN PRIME NUMBER THEORY IGOR E. PRITSKER Abstract. The original Gelfond-Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on =-=[0, 1] t-=-o give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev’... |

8 |
Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungs-Kräfte (1839). Werke 5
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- 1867
(Show Context)
Citation Context ..., i = 1, . . . , k. We develop the ideas of [21] and [7], and establish a connection with the weighted potential theory (or potential theory with external fields) that originated in the work of Gauss =-=[14]-=- and Frostman [13] (see [25] for a modern account on this theory). An important part of the method is the analysis of the asymptotic behavior for the supremum norms of the weighted Vandermonde determi... |

8 | Small polynomials with integer coefficients
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Citation Context ... Chebyshev constant (1.8) tZ([0, 1]) := lim n→∞ � inf 0�≡pn∈Zn[x] �pn� [0,1] � 1/n . It was found by Gorshkov [15] in 1956 that (1.7) can never be achieved. In fact, 0.4213 < tZ([0, =-=1]) < 0.4232 (see [22]-=- for a survey of recent results on this problem). Thus the Gelfond-Schnirelman method failed in its original form, but one can generalize it for polynomials in many variables. Such an idea apparently ... |

8 | Discrepancy of signed measures and polynomial approximation - Andrievskii, Blatt - 2002 |

8 |
An elementary proof of the prime-number theorem
- Selberg
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Citation Context ...hing that ζ(s) does not have zeros on the line {1 + it, t ∈ R}. But the “elementary” approaches to the PNT, which do not use complex analysis and the zeta function, still remained attractive. Selberg =-=[26]-=- and Erdős [11] found the first elementary proof of the Prime Number Theorem in 1949. A survey of elementary methods, with detailed history, may be found in Diamond [10]. The subject of this paper is ... |

7 |
J.P.: On the multivariate transfinite diameter
- Bloom, Calvi
- 1999
(Show Context)
Citation Context ... other sequences of polynomials providing good bounds for the arithmetic functions, along the discussed lines. Among the natural candidates are the multivariate Vandermonde determinants (see [32] and =-=[3]-=-) and other sequences of minimal polynomials [4]. This subject is closely related to pluripotential theory [18]. 2. Potential theory with external fields We consider a special case of the weighted ene... |

7 |
Approximation of functions with Diophantine conditions by polynomials with integral coefficients
- Trigub
- 1971
(Show Context)
Citation Context ...ts on this problem). Thus the Gelfond-Schnirelman method failed in its original form, but one can generalize it for polynomials in many variables. Such an idea apparently had first appeared in Trigub =-=[31], and was-=- independently implemented by Nair [21] and Chudnovsky [7]. The basis of their argument lies in another equivalent form of the Prime Number Theorem [17]: (1.9) � x 1 ψ(t) dt ∼ x2 2 as x → +∞.... |

5 |
A new method in elementary prime number theory
- NAIR
- 1982
(Show Context)
Citation Context ...n method failed in its original form, but one can generalize it for polynomials in many variables. Such an idea apparently had first appeared in Trigub [31], and was independently implemented by Nair =-=[21] and Chud-=-novsky [7]. The basis of their argument lies in another equivalent form of the Prime Number Theorem [17]: (1.9) � x 1 ψ(t) dt ∼ x2 2 as x → +∞. k=0sTHE GELFOND-SCHNIRELMAN METHOD IN PRIME NUM... |

4 |
On multivariate minimal polynomials
- Bloom, Calvi
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(Show Context)
Citation Context ...ounds for the arithmetic functions, along the discussed lines. Among the natural candidates are the multivariate Vandermonde determinants (see [32] and [3]) and other sequences of minimal polynomials =-=[4]-=-. This subject is closely related to pluripotential theory [18]. 2. Potential theory with external fields We consider a special case of the weighted energy problem on a segment of the real line [a, b]... |

4 |
On the distance from zero on the interval [0, 1] of polynomials with integral coefficients
- Gorshkov
- 1959
(Show Context)
Citation Context ...er integers, of degree at most n. In view of (1.6)-(1.7), we are interested in the integer Chebyshev constant (1.8) tZ([0, 1]) := lim n→∞ � inf 0�≡pn∈Zn[x] �pn� [0,1] � 1/n . It was =-=found by Gorshkov [15]-=- in 1956 that (1.7) can never be achieved. In fact, 0.4213 < tZ([0, 1]) < 0.4232 (see [22] for a survey of recent results on this problem). Thus the Gelfond-Schnirelman method failed in its original f... |

4 | Potential and discrepancy estimates for weighted extremal points, Constructive Approximation 16
- Götz, Saff
- 2000
(Show Context)
Citation Context ...measure for (2.2)-(2.3), and (2.17) holds true (cf. Section III.1 of [25]). Thus the discrete problem is a good approximation of the continuous one. We deduce (2.18) from the results of Götz and Saff=-= [16].-=- They require that log w(x) be Hölder continuous on [a, b], which is not true if w of (2.1) has zeros on [a, b]. But we can modify w in the small neighborhoods of those zeros, outside the compact set... |

3 |
Pluripotential theory, Oxford Univ
- Klimek
- 1991
(Show Context)
Citation Context ... Among the natural candidates are the multivariate Vandermonde determinants (see [32] and [3]) and other sequences of minimal polynomials [4]. This subject is closely related to pluripotential theory =-=[18]. 2.-=- Potential theory with external fields We consider a special case of the weighted energy problem on a segment of the real line [a, b], which is associated with the “polynomial-type” weights (1.12)... |

2 |
méthode de variation de Gauss et les fonctions sousharmoniques
- Frostman, La
(Show Context)
Citation Context .... We develop the ideas of [21] and [7], and establish a connection with the weighted potential theory (or potential theory with external fields) that originated in the work of Gauss [14] and Frostman =-=[13]-=- (see [25] for a modern account on this theory). An important part of the method is the analysis of the asymptotic behavior for the supremum norms of the weighted Vandermonde determinants (1.10), whic... |

2 |
On the multivariate transfinite
- Bloom, Calvi
- 1999
(Show Context)
Citation Context ...to design other sequences of polynomials providing good bounds for the arithmetic functions, along the lines discussed. Among the natural candidates are the multivariate Vandermonde determinants (see =-=[3, 32]-=-) and other sequences of minimal polynomials [4]. This subject is closely related to pluripotential theory [18]. 2 Potential Theory With External Fields We consider a special case of the weighted ener... |

2 |
Transfinite diameter, Čebyˇsev constants, and capacity for compacta in C n
- Zaharjuta
(Show Context)
Citation Context ...to design other sequences of polynomials providing good bounds for the arithmetic functions, along the lines discussed. Among the natural candidates are the multivariate Vandermonde determinants (see =-=[3, 32]-=-) and other sequences of minimal polynomials [4]. This subject is closely related to pluripotential theory [18]. 2 Potential Theory With External Fields We consider a special case of the weighted ener... |

1 |
multivariate minimal polynomials
- On
(Show Context)
Citation Context ...ood bounds for the arithmetic functions, along the lines discussed. Among the natural candidates are the multivariate Vandermonde determinants (see [3, 32]) and other sequences of minimal polynomials =-=[4]-=-. This subject is closely related to pluripotential theory [18]. 2 Potential Theory With External Fields We consider a special case of the weighted energy problem on a segment of the real line [a, b] ... |