Journal articles on the topic 'Number theory – Multiplicative number theory – Primes in progressions'

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1

Koukoulopoulos, Dimitris. "Pretentious multiplicative functions and the prime number theorem for arithmetic progressions." Compositio Mathematica 149, no. 7 (2013): 1129–49. http://dx.doi.org/10.1112/s0010437x12000802.

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AbstractBuilding on the concept ofpretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
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2

Granville, Andrew, Adam J. Harper, and K. Soundararajan. "A new proof of Halász’s theorem, and its consequences." Compositio Mathematica 155, no. 1 (2018): 126–63. http://dx.doi.org/10.1112/s0010437x18007522.

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Halász’s theorem gives an upper bound for the mean value of a multiplicative function$f$. The bound is sharp for general such$f$, and, in particular, it implies that a multiplicative function with$|f(n)|\leqslant 1$has either mean value$0$, or is ‘close to’$n^{it}$for some fixed$t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).
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3

Hammonds, Trajan, Casimir Kothari, Noah Luntzlara, Steven J. Miller, Jesse Thorner, and Hunter Wieman. "The explicit Sato–Tate conjecture for primes in arithmetic progressions." International Journal of Number Theory 17, no. 08 (2021): 1905–23. http://dx.doi.org/10.1142/s179304212150069x.

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Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula: see text], then [Formula: see text] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions.
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4

Puchta, Jan-Christoph. "Primes in short arithmetic progressions." Acta Arithmetica 106, no. 2 (2003): 143–49. http://dx.doi.org/10.4064/aa106-2-4.

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5

BAIER, STEPHAN, and LIANGYI ZHAO. "ON PRIMES IN QUADRATIC PROGRESSIONS." International Journal of Number Theory 05, no. 06 (2009): 1017–35. http://dx.doi.org/10.1142/s1793042109002523.

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6

Ramaré, Olivier, Priyamvad Srivastav, and Oriol Serra. "Product of primes in arithmetic progressions." International Journal of Number Theory 16, no. 04 (2019): 747–66. http://dx.doi.org/10.1142/s1793042120500384.

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We prove that, for all [Formula: see text] and for all invertible residue classes [Formula: see text] modulo [Formula: see text], there exists a natural number [Formula: see text] that is congruent to [Formula: see text] modulo [Formula: see text] and that is the product of exactly three primes, all of which are below [Formula: see text]. The proof is further supplemented with a self-contained proof of the special case of the Kneser Theorem we use.
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7

Wójcik, J. "On a problem in algebraic number theory." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (1996): 191–200. http://dx.doi.org/10.1017/s0305004100074090.

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Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’
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8

Zhou, Binbin. "The Chen primes contain arbitrarily long arithmetic progressions." Acta Arithmetica 138, no. 4 (2009): 301–15. http://dx.doi.org/10.4064/aa138-4-1.

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9

Dudek, Adrian W., Loïc Grenié, and Giuseppe Molteni. "Explicit short intervals for primes in arithmetic progressions on GRH." International Journal of Number Theory 15, no. 04 (2019): 825–62. http://dx.doi.org/10.1142/s1793042119500441.

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10

TANNER, NOAM. "STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS." International Journal of Number Theory 05, no. 01 (2009): 81–88. http://dx.doi.org/10.1142/s1793042109001918.

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In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring 𝔽q[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D.
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11

Saradha, N., and R. Tijdeman. "Arithmetic progressions with common difference divisible by small primes." Acta Arithmetica 131, no. 3 (2008): 267–79. http://dx.doi.org/10.4064/aa131-3-4.

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12

Debaene, Korneel. "Explicit counting of ideals and a Brun–Titchmarsh inequality for the Chebotarev density theorem." International Journal of Number Theory 15, no. 05 (2019): 883–905. http://dx.doi.org/10.1142/s1793042119500477.

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We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun–Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg’s Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to [Formula: see text].
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13

Park, Poo-Sung. "Additive uniqueness of PRIMES − 1 for multiplicative functions." International Journal of Number Theory 16, no. 06 (2020): 1369–76. http://dx.doi.org/10.1142/s1793042120500724.

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Let [Formula: see text] be the set of all primes. A function [Formula: see text] is called multiplicative if [Formula: see text] and [Formula: see text] when [Formula: see text]. We show that a multiplicative function [Formula: see text] which satisfies [Formula: see text] satisfies one of the following: (1) [Formula: see text] is the identity function, (2) [Formula: see text] is the constant function with [Formula: see text], (3) [Formula: see text] for [Formula: see text] unless [Formula: see text] is odd and squareful. As a consequence, a multiplicative function which satisfies [Formula: see text] is the identity function.
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14

Bachman, Gennady. "Some Remarks on Nonnegative Multiplicative Functions on Arithmetic Progressions." Journal of Number Theory 73, no. 1 (1998): 72–91. http://dx.doi.org/10.1006/jnth.1998.2279.

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15

Elliott, P. D. T. A. "Multiplicative Functions on Arithmetic Progressions. VI. More Middle Moduli." Journal of Number Theory 44, no. 2 (1993): 178–208. http://dx.doi.org/10.1006/jnth.1993.1044.

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16

Elliott, P. D. T. A. "Primes in progressions to moduli with a large power factor." Ramanujan Journal 13, no. 1-3 (2007): 241–51. http://dx.doi.org/10.1007/s11139-006-0250-4.

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17

RIBENBOIM, PAULO. "MULTIPLE PATTERNS OF k-TUPLES OF INTEGERS." International Journal of Number Theory 07, no. 07 (2011): 1761–79. http://dx.doi.org/10.1142/s1793042111004733.

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The first proposition and its corollary are a transfiguration of Dirichlet's pigeon-hole principle. They are applied to show that a wide variety of sequences display arbitrarily large patterns of sums, differences, higher differences, etc. Among these, we include sequences of primes in arithmetic progressions, of powerful integers, sequences of integers with radical index having a prescribed lower bound, and many others. We also deal with patterns in iterated sequences of primes, patterns of gaps between primes, patterns of values of Euler's φ-function, or their gaps, as well as patterns related to the sequence of Carmichael numbers.
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18

KNAFO, EMMANUEL. "EFFECTIVE LOWER BOUND FOR THE VARIANCE OF DISTRIBUTION OF PRIMES IN ARITHMETIC PROGRESSIONS." International Journal of Number Theory 04, no. 01 (2008): 45–56. http://dx.doi.org/10.1142/s1793042108001213.

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Through a refinement for the estimation of the effect of Siegel zeros, we show how to avoid the use of Siegel's theorem in order to obtain the first effective lower bound for the variance of distribution of primes in arithmetic progressions.
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19

Chirre, Andrés, Valdir Pereira Júnior, and David de Laat. "Primes in arithmetic progressions and semidefinite programming." Mathematics of Computation 90, no. 331 (2021): 2235–46. http://dx.doi.org/10.1090/mcom/3638.

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Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math. Helv. 94, no. 3 (2019)]. For this we extend the Guinand-Weil explicit formula over all Dirichlet characters modulo q ≥ 3 q \geq 3 , and we reduce the associated extremal problems to convex optimization problems that can be solved numerically via semidefinite programming.
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20

Ramaré, Olivier, and Aled Walker. "Products of primes in arithmetic progressions: a footnote in parity breaking." Journal de Théorie des Nombres de Bordeaux 30, no. 1 (2018): 219–25. http://dx.doi.org/10.5802/jtnb.1024.

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21

Halupczok, Karin. "Goldbach’s problem with primes in arithmetic progressions and in short intervals." Journal de Théorie des Nombres de Bordeaux 25, no. 2 (2013): 331–51. http://dx.doi.org/10.5802/jtnb.839.

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22

Aaccagnini, Alessandro. "A note on large gaps between consecutive primes in arithmetic progressions." Journal of Number Theory 42, no. 1 (1992): 100–102. http://dx.doi.org/10.1016/0022-314x(92)90111-2.

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23

Elliott, P. "Products of shifted primes: Multiplicative analogues of Goldbach's problem." Acta Arithmetica 88, no. 1 (1999): 31–50. http://dx.doi.org/10.4064/aa-88-1-31-50.

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24

Bauer, Claus, and Yonghui Wang. "The binary Goldbach conjecture with primes in arithmetic progressions with large modulus." Acta Arithmetica 159, no. 3 (2013): 227–43. http://dx.doi.org/10.4064/aa159-3-2.

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25

Schinzel, A., and J. Wójcik. "On a problem in elementary number theory." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 2 (1992): 225–32. http://dx.doi.org/10.1017/s0305004100070912.

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Let ordq (a) be the exponent with which a prime q occurs in the factorization of a rational number a ≠ 0 and, if ordq (a) = 0, let Mq(a) be the multiplicative group generated by a modulo q. In the course of a group-theoretical investigation J. S. Wilson found he needed some results about integers a, b such that Mq(a) = Mq(b), indeed also for algebraic integers, and he proved some of what he needed. J. W. S. Cassels observed that Wilson's argument naturally proved the existence of infinitely many primes q with Mq(a) = Mq(b) for rational integers a, b with ab > 0, |a| > 1, |b| > 1. J. G. Thompson found a proof for the case of integers a, b with ab < 0, |a| > 1, |b| > 1. He also posed the problem for rational a, b (all this is unpublished). The aim of this paper is to prove that the answer to Thompson's question is affirmative. We also include the case ab > 0 settled by Thompson himself. We use the same technique devised by Wilson which has been elaborated by Cassels and Thompson. We thank Professor Cassels for the simplification of our original exposition and the referee for his suggestions.
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26

Maier, Helmut, and Saurabh Kumar Singh. "On multiplicative functions with mean-value close to −1 on primes." Journal of Number Theory 197 (April 2019): 89–105. http://dx.doi.org/10.1016/j.jnt.2018.08.001.

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27

Pritchard, Paul A. "Long arithmetic progressions of primes: some old, some new." Mathematics of Computation 45, no. 171 (1985): 263. http://dx.doi.org/10.1090/s0025-5718-1985-0790659-1.

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28

Halupczok, Karin. "On the ternary Goldbach problem with primes in arithmetic progressions having a common modulus." Journal de Théorie des Nombres de Bordeaux 21, no. 1 (2009): 203–13. http://dx.doi.org/10.5802/jtnb.666.

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29

Dudek, Adrian W., and David J. Platt. "On the sum of the square of a prime and a square-free number." LMS Journal of Computation and Mathematics 19, no. 1 (2016): 16–24. http://dx.doi.org/10.1112/s1461157015000297.

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We prove that every integer$n\geqslant 10$such that$n\not \equiv 1\text{ mod }4$can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erdős that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author’s numerical computation regarding the generalised Riemann hypothesis to extend the explicit bounds of Ramaré–Rumely.
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30

Guiasu, Silviu. "Detecting a Regularity in the Generation and Utilization of Primes in the Multiplicative Number Theory." Natural Science 11, no. 06 (2019): 187–96. http://dx.doi.org/10.4236/ns.2019.116019.

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31

LUCA, FLORIAN, and YOHEI TACHIYA. "IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE." International Journal of Number Theory 10, no. 03 (2014): 623–36. http://dx.doi.org/10.1142/s1793042113501121.

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Let q be an integer with |q| > 1 and {an}n≥1 be an eventually periodic sequence of rational numbers, not identically zero from some point on. Then the number [Formula: see text] is irrational. In particular, if the periodic sequences [Formula: see text] of rational numbers are linearly independent over ℚ, then so are the following m + 1 numbers: [Formula: see text] This generalizes a result of Erdős who treated the case of m = 1 and [Formula: see text]. The method of proof is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.
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32

Hudson, Richard H. "Averaging effects on irregularities in the distribution of primes in arithmetic progressions." Mathematics of Computation 44, no. 170 (1985): 561. http://dx.doi.org/10.1090/s0025-5718-1985-0777286-7.

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33

SMILANSKY, YOTAM. "SUMS OF TWO SQUARES — PAIR CORRELATION AND DISTRIBUTION IN SHORT INTERVALS." International Journal of Number Theory 09, no. 07 (2013): 1687–711. http://dx.doi.org/10.1142/s1793042113500516.

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In this work we show that based on a conjecture for the pair correlation of integers representable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the number of representable integers in short intervals is consistent with a Poisson distribution, where "short" means of length comparable to the mean spacing between sums of two squares. In addition we present a method for producing such conjectures through calculations in prime power residue rings and describe how these conjectures, as well as the above stated result, may by generalized to other binary quadratic forms. While producing these pair correlation conjectures we arrive at a surprising result regarding Mertens' formula for primes in arithmetic progressions, and in order to test the validity of the conjectures, we present numerical computations which support our approach.
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34

Kadiri, H. "Short effective intervals containing primes in arithmetic progressions and the seven cubes problem." Mathematics of Computation 77, no. 263 (2008): 1733–48. http://dx.doi.org/10.1090/s0025-5718-08-02084-x.

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35

Lebowitz-Lockard, Noah. "Additively unique sets of prime numbers." International Journal of Number Theory 14, no. 10 (2018): 2757–65. http://dx.doi.org/10.1142/s179304211850166x.

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Spiro proved that the identity function is the only multiplicative function with [Formula: see text] for some prime [Formula: see text] and [Formula: see text] for all prime [Formula: see text] and [Formula: see text]. We determine the sets [Formula: see text] of primes for which restricting our condition to [Formula: see text] for all [Formula: see text] still implies that [Formula: see text] is the identity function. We prove that [Formula: see text] satisfies these conditions if and only if [Formula: see text] contains every prime that is not the larger element of a twin prime pair and [Formula: see text] contains [Formula: see text] or [Formula: see text].
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36

Hall, R. R., and G. Tenenbaum. "Effective mean value estimates for complex multiplicative functions." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 2 (1991): 337–51. http://dx.doi.org/10.1017/s0305004100070419.

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Quantitative estimates for finite mean valuesof multiplicative functions are highly applicable tools in analytic and probabilistic number theory. Extending a result of Hall [4], Halberstam and Richert[3] proved a useful inequality valid for real, non-negative g satisfying for instance a Wirsing type condition, viz for all primes p, with constants λ1 ≥ 0, 0 ≤ λ2 < 2. Their upper bound is sharp to within a factor (l + o(l)), but even a weaker and easier to prove estimate, such as(where the implied constants depend on λ1 and λ2), may become a surprisingly strong device. For instance, setting g(p) = l ± ε, where ε is an arbitrarily small positive number, provides immediately a proof of the famous Hardy–Ramanujan theorem on the normal order of the number of prime factors of an integer. This example, and many others, are discussed in detail in our book [5] where we make extensive use of (2) for various problems connected with the structure of the set of divisors of a normal number.
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37

KACZOROWSKI, J., and A. PERELLI. "On the distribution in short intervals of products of a prime and integers from a given set." Mathematical Proceedings of the Cambridge Philosophical Society 124, no. 1 (1998): 1–14. http://dx.doi.org/10.1017/s0305004197002284.

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A classical problem in analytic number theory is the distribution in short intervals of integers with a prescribed multiplicative structure, such as primes, almost-primes, k-free numbers and others. Recently, partly due to applications to cryptology, much attention has been received by the problem of the distribution in short intervals of integers without large prime factors, see Lenstra–Pila–Pomerance [3] and section 5 of the excellent survey by Hildebrand–Tenenbaum [1].In this paper we deal with the distribution in short intervals of numbers representable as a product of a prime and integers from a given set [Sscr ], defined in terms of cardinality properties. Our results can be regarded as an extension of the above quoted results, and we will provide a comparison with such results by a specialization of the set [Sscr ].
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38

Tumanova, Elena Alexandrovna. "Computational Analysis of Quantitative Characteristics of some Residual Properties of Solvable Baumslag-Solitar Groups." Modeling and Analysis of Information Systems 28, no. 2 (2021): 136–45. http://dx.doi.org/10.18255/1818-1015-2021-2-136-145.

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Let $G_{k}$ be defined as $G_{k} = \langle a, b;\ a^{-1}ba = b^{k} \rangle$, where $k \ne 0$. It is known that, if $p$ is some prime number, then $G_{k}$ is residually a finite $p$-group if and only if $p \mid k - 1$. It is also known that, if $p$ and $q$ are primes not dividing $k - 1$, $p < q$, and $\pi = \{p,\,q\}$, then $G_{k}$ is residually a finite $\pi$-group if and only if $(k, q) = 1$, $p \mid q - 1$, and the order of $k$ in the multiplicative group of the field $\mathbb{Z}_{q}$ is a $p$\-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let $f_{k}(x)$ be the number of sets $\{p,\,q\}$ such that $p < q$, $p \nmid k - 1$, $q \nmid k - 1$, $(k, q) = 1$, $p \mid q - 1$, the order of $k$ modulo $q$ is a $p$\-number, and $p$, $q$ are chosen among the first $x$ primes. We state that, if $2 \leq |k| \leq 10000$ and $1 \leq x \leq 50000$, then, for almost all considered $k$, the function $f_{k}(x)$ can be approximated quite accurately by the function $\alpha_{k}x^{0.85}$, where the coefficient $\alpha_{k}$ is different for each $k$ and $\{\alpha_{k} \mid 2 \leq |k| \leq 10000\} \subseteq (0.28;\,0.31]$. We also investigate the dependence of the value $f_{k}(50000)$ on $k$ and propose an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion. The results obtained may have applications in the theory of computational complexity and algebraic cryptography.
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39

LEWIS, ROBERT R. "Improved Upper Bounds for the Order of Some Classes of Abelian Cayley and Circulant Graphs of Diameter Two." Journal of Interconnection Networks 17, no. 03n04 (2017): 1741012. http://dx.doi.org/10.1142/s0219265917410122.

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In the degree-diameter problem for Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d there is a wide gap between the best lower and upper bounds valid for all d, being quadratic functions with leading coefficient 1/4 and 1/2 respectively. Recent papers have presented constructions which increase the coefficient of the lower bound to be at or just below 3/8 for sparse sets of degree d related to primes of specific congruence classes. The constructions use the direct product of the multiplicative and additive subgroups of a Galois field and a specific cyclic group of coprime order. It was anticipated that this approach would be capable of yielding further improvement towards the upper bound value of 1/2. In this paper, however, it is proved that the quadratic coefficient of the order of families of Abelian Cayley graphs of this class of construction can never exceed the value of 3/8, establishing an asymptotic limit of 3/8 for the quadratic coefficient of families of extremal graphs of this class. By applying recent results from number theory these constructions can be extended to be valid for every degree above some threshold, establishing an improved asymptotic lower bound approaching 3/8 for general Abelian Cayley and circulant graphs of diameter 2 and arbitrary degree d.
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40

Sachpazis, Stelios. "Multiplicative Functions on Shifted Primes." Journal of Number Theory, July 2021. http://dx.doi.org/10.1016/j.jnt.2021.06.027.

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41

Kaptan, Deniz A. "A note on small gaps between primes in arithmetic progressions." Acta Arithmetica, 2016, 1–25. http://dx.doi.org/10.4064/aa8277-11-2015.

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42

Mastrostefano, Daniele. "A lower bound for the variance of generalized divisor functions in arithmetic progressions." Ramanujan Journal, March 12, 2021. http://dx.doi.org/10.1007/s11139-020-00374-8.

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AbstractWe prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $$\alpha $$ α -fold divisor function, for any complex number $$\alpha \not \in \{1\}\cup -\mathbb {N}$$ α ∉ { 1 } ∪ - N , even when considering a sequence of parameters $$\alpha $$ α close in a proper way to 1. Our work builds on that of Harper and Soundararajan, who handled the particular case of k-fold divisor functions $$d_k(n)$$ d k ( n ) , with $$k\in \mathbb {N}_{\ge 2}$$ k ∈ N ≥ 2 .
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43

Alabdali, Ali A., and Nigel P. Byott. "Skew braces of squarefree order." Journal of Algebra and Its Applications, July 17, 2020, 2150128. http://dx.doi.org/10.1142/s0219498821501280.

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Let [Formula: see text] be a squarefree integer, and let [Formula: see text], [Formula: see text] be two groups of order [Formula: see text]. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group [Formula: see text] and additive group [Formula: see text]. As an application, we enumerate skew braces whose order is the product of three distinct primes, in particular proving a conjecture of Bardakov, Neshchadim and Yadav on the number of skew braces of order [Formula: see text] for primes [Formula: see text].
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44

Hegarty, Peter, and Anders Martinsson. "Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups." Electronic Journal of Combinatorics 22, no. 4 (2015). http://dx.doi.org/10.37236/5340.

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A permutation $\pi$ of an abelian group $G$ is said to destroy arithmetic progressions (APs) if, whenever $(a, \, b, \, c)$ is a non-trivial 3-term AP in $G$, that is $c-b=b-a$ and $a, \, b, \, c$ are not all equal, then $(\pi(a), \, \pi(b), \pi(c))$ is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of $\mathbb{Z}_n$, for all $n \not\in \{2, \, 3, \, 5, \, 7\}$. Here we prove, as a special case of a more general result, that such a permutation exists for all $n \geq n_0$, for some explicitly constructed number $n_0 \approx 1.4 \times 10^{14}$. We also construct such a permutation of $\mathbb{Z}_p$ for all primes $p > 3$ such that $p \equiv 3 \; ({\hbox{mod $8$}})$.
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45

Goldfeld, Dorian, Eric Stade, and Michael Woodbury. "An orthogonality relation for (with an appendix by Bingrong Huang)." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.39.

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Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.
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46

Wu, Tingting, Shixin Zhu, Li Liu, and Lanqiang Li. "Repeated-root constacyclic codes of length 6lmpn." Advances in Mathematics of Communications, 2021, 0. http://dx.doi.org/10.3934/amc.2021044.

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document}</tex-math></inline-formula> is decomposed into a mutually disjoint union of <inline-formula><tex-math id="M4">\begin{document}$ \gcd(6l^mp^n,q-1) $\end{document}</tex-math></inline-formula> cosets over subgroup <inline-formula><tex-math id="M5">\begin{document}$ <\xi^{6l^mp^n}> $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a primitive element of <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>. Based on the decomposition, the structure of constacyclic codes of length <inline-formula><tex-math id="M8">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over finite field <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> and their duals is established in terms of their generator polynomials, where <inline-formula><tex-math id="M10">\begin{document}$ p\neq{3} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ l\neq{3} $\end{document}</tex-math></inline-formula> are distinct odd primes, <inline-formula><tex-math id="M12">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>.</p>
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