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Greg Martin - One of the best experts on this subject based on the ideXlab platform.
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Inequities in the Shanks-Renyi Prime Number Race: An Asymptotic Formula for the densities
2020Co-Authors: Daniel Fiorilli, Greg MartinAbstract:Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if $a$ is a nonsquare\mod q and $b$ is a square\mod q, then there tend to be more primes congruent to $a\mod q$ than $b\mod q$ in initial intervals of the positive integers; more succinctly, there is a tendency for $\pi(x;q,a)$ to exceed $\pi(x;q,b)$. Rubinstein and Sarnak defined $\delta(q;a,b)$ to be the logarithmic density of the set of positive real numbers $x$ for which this inequality holds; intuitively, $\delta(q;a,b)$ is the "probability" that $\pi(x;q,a) > \pi(x;q,b)$ when $x$ is "chosen randomly". In this paper, we establish an Asymptotic series for $\delta(q;a,b)$ that can be instantiated with an error term smaller than any negative power of $q$. This Asymptotic Formula is written in terms of a variance $V(q;a,b)$ that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet $L$-functions corresponding to characters\mod q; we show how $V(q;a,b)$ can be evaluated exactly as a finite expression. In addition to providing the exact rate at which $\delta(q;a,b)$ converges to $\frac12$ as $q$ grows, these evaluations allow us to compare the various density values $\delta(q;a,b)$ as $a$ and $b$ vary modulo $q$; by analyzing the resulting Formulas, we can explain and predict which of these densities will be larger or smaller, based on arithmetic properties of the residue classes $a$ and $b\mod q$. For example, we show that if $a$ is a prime power and $a'$ is not, then $\delta(q;a,1) < \delta(q;a',1)$ for all but finitely many moduli $q$ for which both $a$ and $a'$ are nonsquares. Finally, we establish rigorous numerical bounds for these densities $\delta(q;a,b)$ and report on extensive calculations of them.
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inequities in the shanks renyi prime number race an Asymptotic Formula for the densities
Crelle's Journal, 2013Co-Authors: Daniel Fiorilli, Greg MartinAbstract:Chebyshev was the first to observe a bias in the distribution of primes in residue classes. The general phenomenon is that if a is a nonsquare ðmod qÞ and b is a square ðmod qÞ, then there tend to be more primes congruent to a ðmod qÞ than b ðmod qÞ in initial intervals of the positive integers; more succinctly, there is a tendency for pðx; q; aÞ to exceed pðx; q; bÞ. Rubinstein and Sarnak defined dðq; a; bÞ to be the logarithmic density of the set of positive real numbers x for which this inequality holds; intuitively, dðq; a; bÞ is the ''probability'' that pðx; q; aÞ > pðx; q; bÞ when x is ''chosen randomly''. In this paper, we establish an Asymptotic series for dðq; a; bÞ that can be instantiated with an error term smaller than any negative power of q. This Asymptotic Formula is written in terms of a variance V ðq; a; bÞ that is originally defined as an infinite sum over all nontrivial zeros of Dirichlet L-functions corresponding to characters ðmod qÞ; we show how V ðq; a; bÞ can be evaluated exactly as a finite expression. In addition to providing the exact rate at which dðq; a; bÞ converges to 1= 2a sq grows, these evaluations allow us to compare the various density values dðq; a; bÞ as a and b vary modulo q; by analyzing the resulting Formulas, we can explain and predict which of these densities will be larger or smaller, based on arithme- tic properties of the residue classes a and b ðmod qÞ. For example, we show that if a is a prime power and a 0 is not, then dðq; a; 1Þ < dðq; a 0 ; 1Þ for all but finitely many moduli q for which both a and a 0 are nonsquares. Finally, we establish rigorous numerical bounds for these densities dðq; a; bÞ and report on extensive calculations of them, including for ex- ample the determination of all 117 density values that exceed 9=10.
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An Asymptotic Formula for the Number of Smooth Values of a Polynomial
arXiv: Number Theory, 1999Co-Authors: Greg MartinAbstract:Although we expect to find many smooth numbers (i.e., numbers with no large prime factors) among the values taken by a polynomial with integer coefficients, it is unclear what the Asymptotic number of such smooth values should be; this is in contrast to the related problem of counting the number of prime values of a polynomial, for which Bateman and Horn published a conjectured Asymptotic Formula that is widely believed to be true. We discuss how to employ the Bateman-Horn conjecture to derive an Asymptotic Formula for the number of smooth values of a polynomial, with the smoothness parameter in a non-trivial range. This conditional result provides a believable heuristic for the number of smooth integers among all values {F(n)}, and also among the values {F(p)} on prime arguments only.
A Morozov - One of the best experts on this subject based on the ideXlab platform.
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zamolodchikov Asymptotic Formula and instanton expansion in 2 susy nf 2nc qcd
Journal of High Energy Physics, 2009Co-Authors: A Marshakov, A Mironov, A MorozovAbstract:The AGT relations allow to convert the Zamolodchikov Asymptotic Formula for conformal block into the instanton expansion of the Seiberg-Witten prepotential for the theory with two colors and four fundamental flavors. This provides an explicit Formula for the instanton corrections in this model. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory q0 = eiπτ0 plays the role of a branch point on the spectral elliptic curve. The exact prepotential at this point is = (1/2πi)a2log q with q≠q0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q0 = θ104/θ004(q) = 16q(1+O(q)), i.e. the Zamolodchikov Asymptotic Formula gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.
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zamolodchikov Asymptotic Formula and instanton expansion in n 2 susy n_f 2n_c qcd
arXiv: High Energy Physics - Theory, 2009Co-Authors: A Marshakov, A Mironov, A MorozovAbstract:The AGT relations allow one to convert the Zamolodchikov Asymptotic Formula for the conformal block into the instanton expansion of the Seiberg-Witten prepotential for theory with two colors and four fundamental flavors. This provides an explicit Formula for the instanton corrections in this model, resolving in this way an old problem in Seiberg-Witten theory. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory 16q_0 = 16e^{i\pi\tau_0} plays the role of a branch point on the spectral torus. The exact prepotential at this point is F a^2\log q with q\neq q_0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q_0 = \theta_{10}^4/\theta_{00}^4(q) = 16q(1+O(q)), i.e. the Zamolodchikov Asymptotics gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.
A Marshakov - One of the best experts on this subject based on the ideXlab platform.
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zamolodchikov Asymptotic Formula and instanton expansion in 2 susy nf 2nc qcd
Journal of High Energy Physics, 2009Co-Authors: A Marshakov, A Mironov, A MorozovAbstract:The AGT relations allow to convert the Zamolodchikov Asymptotic Formula for conformal block into the instanton expansion of the Seiberg-Witten prepotential for the theory with two colors and four fundamental flavors. This provides an explicit Formula for the instanton corrections in this model. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory q0 = eiπτ0 plays the role of a branch point on the spectral elliptic curve. The exact prepotential at this point is = (1/2πi)a2log q with q≠q0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q0 = θ104/θ004(q) = 16q(1+O(q)), i.e. the Zamolodchikov Asymptotic Formula gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.
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zamolodchikov Asymptotic Formula and instanton expansion in n 2 susy n_f 2n_c qcd
arXiv: High Energy Physics - Theory, 2009Co-Authors: A Marshakov, A Mironov, A MorozovAbstract:The AGT relations allow one to convert the Zamolodchikov Asymptotic Formula for the conformal block into the instanton expansion of the Seiberg-Witten prepotential for theory with two colors and four fundamental flavors. This provides an explicit Formula for the instanton corrections in this model, resolving in this way an old problem in Seiberg-Witten theory. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory 16q_0 = 16e^{i\pi\tau_0} plays the role of a branch point on the spectral torus. The exact prepotential at this point is F a^2\log q with q\neq q_0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q_0 = \theta_{10}^4/\theta_{00}^4(q) = 16q(1+O(q)), i.e. the Zamolodchikov Asymptotics gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.
Jan Moser - One of the best experts on this subject based on the ideXlab platform.
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jacob s ladders and the Asymptotic Formula for the integral of the eight order expression zeta 1 2 i vp_2 t 4 zeta 1 2 it 4
arXiv: Classical Analysis and ODEs, 2010Co-Authors: Jan MoserAbstract:It is proved in this paper that there is a fine correlation between the values of $|\zeta(1/2+i\vp_2(t))|^4$ and $|\zeta(1/2+it)|^4$ where $\vp_2(t)$ stands for the Jacob's ladder of the second order. This new Asymptotic Formula cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
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jacob s ladders and the first Asymptotic Formula for the expression of the fifth order z 3 hat z 2 t for the collection of disconnected sets
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Jan MoserAbstract:It is shown in this paper that there is a fine correlation of the fifth order between the values $Z[\phi(t)/2+\rho_1]Z[\phi(t)/2+\rho_2]Z[\phi(t)/2+\rho_3]$ and $\hat{Z}^2(t)$ which correspond to two collections of disconnected sets. This new Asymptotic Formula cannot be obtained within known theories of Balasubramanian, Heath-Brown and Ivic.
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jacob s ladders and the first Asymptotic Formula for the expression of the sixth order zeta 1 2 i varphi t 2 4 zeta 1 2 it 2
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Jan MoserAbstract:t is proved in this paper that there is a fine correlation between the values of $|\zeta(1/2+i\varphi(t)/2)|^4$ and $|\zeta(1/2+it)|^2$ which correspond to two segments with gigantic distance each from other. This new Asymptotic Formula cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
Dongxue Zhou - One of the best experts on this subject based on the ideXlab platform.
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Asymptotic Formula on average path length of a special network based on sierpinski carpet
Fractals, 2018Co-Authors: Dongxue ZhouAbstract:In this paper, we construct a special network based on the construction of the Sierpinski carpet. Using the self-similarity and renewal theorem, we obtain the Asymptotic Formula for the average pat...
