scholarly journals Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

2020 ◽  
Vol 63 (4) ◽  
pp. 837-849 ◽  
Author(s):  
Lucile Devin
Keyword(s):  
Riemann Hypothesis ◽  
Linear Independence ◽  
Classical Case ◽  
Weak Version ◽  
Distribution Of Primes ◽  
Logarithmic Density ◽  
Fixed Integer ◽  
Residue Classes ◽  
Independence Hypothesis ◽  
And Function

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

scholarly journals Large deviations of the limiting distribution in the Shanks–Rényi prime number race

2012 ◽  
Vol 153 (1) ◽  
pp. 147-166 ◽  
Author(s):  
YOUNESS LAMZOURI
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Prime Number ◽  
Riemann Hypothesis ◽  
Linear Independence ◽  
Limiting Distribution ◽  
Absolutely Continuous ◽  
Independence Hypothesis ◽  
Vector Valued Function ◽  
Vector Valued

AbstractLet q ≥ 3, 2 ≤ r ≤ φ(q) and a1, . . ., ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function Eq;a1, . . ., ar(x) = (E(x;q,a1), . . ., E(x;q,ar)), where $E(x;q,a)= ({\log x}/{\sqrt{x}})(\phi(q)\pi(x;q,a)-\pi(x))$, has a limiting distribution μq;a1, . . ., ar which is absolutely continuous on $\mathbb{R}^r$. Furthermore, they proved that for r fixed, μq;a1, . . ., ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of μq;a1, . . ., ar.

scholarly journals Variants on Andrica’s Conjecture with and without the Riemann Hypothesis

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 289 ◽  
Author(s):  
Matt Visser
Keyword(s):  
Riemann Hypothesis ◽  
Distribution Of Primes

The gap between what we can explicitly prove regarding the distribution of primes and what we suspect regarding the distribution of primes is enormous. It is (reasonably) well-known that the Riemann hypothesis is not sufficient to prove Andrica’s conjecture: ∀n≥1, is p n + 1 - p n ≤ 1 ? However, can one at least get tolerably close? I shall first show that with a logarithmic modification, provided one assumes the Riemann hypothesis, one has p n + 1 /ln p n + 1 - p n /ln p n < 11/25; (n ≥ 1). Then, by considering more general mth roots, again assuming the Riemann hypothesis, I show that p n + 1 m - p n m < 44/(25 e[m < 2]); (n ≥ 3; m > 2). In counterpoint, if we limit ourselves to what we can currently prove unconditionally, then the only explicit Andrica-like results seem to be variants on the relatively weak results below: ln2 pn + 1 - ln2 pn < 9; ln3 pn + 1 - ln3 pn < 52; ln4 pn + 1 - ln4 pn < 991; (n ≥ 1). I shall also update the region on which Andrica’s conjecture is unconditionally verified.

Erratum: Limiting properties of the distribution of primes in an arbitrarily large number of residue classes

2020 ◽  
pp. 1-4
Author(s):  
Lucile Devin
Keyword(s):  
Linear Relation ◽  
Distribution Of Primes ◽  
Residue Classes

Abstract As pointed out by Alexandre Bailleul, the paper mentioned in the title contains a mistake in Theorem 2.2. The hypothesis on the linear relation of the almost periods is not sufficient. In this note, we fix the problem and its minor consequences on other results in the same paper.

scholarly journals The André–Oort conjecture for Drinfeld modular varieties

2013 ◽  
Vol 149 (4) ◽  
pp. 507-567 ◽  
Author(s):  
Patrik Hubschmid
Keyword(s):  
Riemann Hypothesis ◽  
Classical Case ◽  
Base Field ◽  
Generalized Riemann Hypothesis ◽  
Fixed Set ◽  
Special Points ◽  
Modular Varieties

AbstractWe consider the analogue of the André–Oort conjecture for Drinfeld modular varieties which was formulated by Breuer. We prove this analogue for special points with separable reflex field over the base field by adapting methods which were used by Klingler and Yafaev to prove the André–Oort conjecture under the generalized Riemann hypothesis in the classical case. Our result extends results of Breuer showing the correctness of the analogue for special points lying in a curve and for special points having a certain behaviour at a fixed set of primes.

scholarly journals INTEGRATION AND CELL DECOMPOSITION IN P-MINIMAL STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 1124-1141 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
EVA LEENKNEGT
Keyword(s):  
Cell Decomposition ◽  
Poincaré Series ◽  
Weak Version ◽  
Poincare Series ◽  
Skolem Functions ◽  
Direct Corollary ◽  
Image Position ◽  
And Function ◽  
Constructible Functions

AbstractWe show that the class of ${\cal L}$-constructible functions is closed under integration for any P-minimal expansion of a p-adic field $\left( {K,{\cal L}} \right)$. This generalizes results previously known for semi-algebraic and subanalytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for P-minimal structures, a result which is independent of the existence of Skolem functions. A direct corollary is that Denef’s results on the rationality of Poincaré series hold in any P-minimal expansion of a p-adic field $\left( {K,{\cal L}} \right)$.

scholarly journals Orthogonality of quasi-orthogonal polynomials

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6953-6977 ◽  
Author(s):  
Cleonice Bracciali ◽  
Francisco Marcellán ◽  
Serhan Varma
Keyword(s):  
Orthogonal Polynomials ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Continuous Functions ◽  
Quadrature Formulas ◽  
Algebraic Polynomials ◽  
Fixed Integer ◽  
Recurrence Formulas ◽  
Polynomial Factor ◽  
Necessary And Sufficient

A result of P?lya states that every sequence of quadrature formulas Qn(f) with n nodes and positive Cotes numbers converges to the integral I(f) of a continuous function f provided Qn(f) = I(f) for a space of algebraic polynomials of certain degree that depends on n. The classical case when the algebraic degree of precision is the highest possible is well-known and the quadrature formulas are the Gaussian ones whose nodes coincide with the zeros of the corresponding orthogonal polynomials and the Cotes (Christoffel) numbers are expressed in terms of the so-called kernel polynomials. In many cases it is reasonable to relax the requirement for the highest possible degree of precision in order to gain the possibility to either approximate integrals of more specific continuous functions that contain a polynomial factor or to include additional fixed nodes. The construction of such quadrature processes is related to quasi-orthogonal polynomials. Given a sequence {Pn}n?0 of monic orthogonal polynomials and a fixed integer k, we establish necessary and sufficient conditions so that the quasi-orthogonal polynomials {Qn}n?0 defined by Qn(x) = Pn(x) + ?k-1,i=1 bi,nPn-i(x), n ? 0, with bi,n ? R, and bk-1,n ? 0 for n ? k-1, also constitute a sequence of orthogonal polynomials. Therefore we solve the inverse problem for linearly related orthogonal polynomials. The characterization turns out to be equivalent to some nice recurrence formulas for the coefficients bi,n. We employ these results to establish explicit relations between various types of quadrature rules from the above relations. A number of illustrative examples are provided.

VARIANCE OF DISTRIBUTION OF PRIMES IN RESIDUE CLASSES

1996 ◽  
Vol 47 (3) ◽  
pp. 313-336 ◽  
Author(s):  
J. B. FRIEDLANDER ◽  
D. A. GOLDSTON
Keyword(s):  
Distribution Of Primes ◽  
Residue Classes

On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

2020 ◽  
Author(s):  
Mehdi Hassani ◽  
Mahmoud Marie Marie
Keyword(s):  
Euler Function ◽  
Fixed Integer ◽  
Residue Classes

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).

scholarly journals Théorèmes de type Fouvry–Iwaniec pour les entiers friables

2015 ◽  
Vol 151 (5) ◽  
pp. 828-862 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Recent Work ◽  
Prime Factor ◽  
Arithmetic Progressions ◽  
Dispersion Method ◽  
Fixed Integer ◽  
Residue Classes

An integer $n$ is said to be $y$-friable if its largest prime factor $P^{+}(n)$ is less than $y$. In this paper, it is shown that the $y$-friable integers less than $x$ have a weak exponent of distribution at least $3/5-{\it\varepsilon}$ when $(\log x)^{c}\leqslant x\leqslant x^{1/c}$ for some $c=c({\it\varepsilon})\geqslant 1$, that is to say, they are well distributed in the residue classes of a fixed integer $a$, on average over moduli ${\leqslant}x^{3/5-{\it\varepsilon}}$ for each fixed $a\neq 0$ and ${\it\varepsilon}>0$. We apply this to the estimation of the sum $\sum _{2\leqslant n\leqslant x,P^{+}(n)\leqslant y}{\it\tau}(n-1)$ when $(\log x)^{c}\leqslant y$. This follows and improves on previous work of Fouvry and Tenenbaum. Our proof combines the dispersion method of Linnik in the setting of Bombieri, Fouvry, Friedlander and Iwaniec with recent work of Harper on friable integers in arithmetic progressions.

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