Relevant bibliographies by topics / Number theory – Multiplicative number theory – Primes in progressions

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters

Academic literature on the topic 'Number theory – Multiplicative number theory – Primes in progressions'

Author: Grafiati
Published: 4 June 2021
Last updated: 14 February 2022

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

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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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.
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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:
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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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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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Dissertations / Theses on the topic "Number theory – Multiplicative number theory – Primes in progressions"

1

Fiorilli, Daniel. "Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques." Thèse, 2011. http://hdl.handle.net/1866/8333.

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Le sujet principal de cette thèse est la distribution des nombres premiers dans les progressions arithmétiques, c'est-à-dire des nombres premiers de la forme $qn+a$, avec $a$ et $q$ des entiers fixés et $n=1,2,3,\dots$ La thèse porte aussi sur la comparaison de différentes suites arithmétiques par rapport à leur comportement dans les progressions arithmétiques. Elle est divisée en quatre chapitres et contient trois articles. Le premier chapitre est une invitation à la théorie analytique des nombres, suivie d'une revue des outils qui seront utilisés plus tard. Cette introduction comporte aussi
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Books on the topic "Number theory – Multiplicative number theory – Primes in progressions"

1

Florian, Luca, ed. Analytic number theory: Exploring the anatomy of integers. American Mathematical Society, 2012.

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2

A course in analytic number theory. American Mathematical Society, 2014.

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3

Lectures on the Riemann zeta function. American Mathematical Society, 2014.

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4

Cojocaru, Alina Carmen, Chantal David, and F. Pappalardi. Scholar, a scientific celebration highlighting open lines of arithmetic research: Conference in honour of M. Ram Murty's mathematical legacy on his 60th birthday, October 15-17, 2013, Centre de Recherches Mathematiques, Universite de Montreal, Quebec, Canada. Edited by Murty Maruti Ram editor. American Mathematical Society, 2015.

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5

Shparlinski, Igor E., and David R. Kohel. Frobenius distributions: Lang-Trotter and Sato-Tate conjectures : Winter School on Frobenius Distributions on Curves, February 17-21, 2014 [and] Workshop on Frobenius Distributions on Curves, February 24-28, 2014, Centre International de Rencontres Mathematiques, Marseille, France. American Mathematical Society, 2016.

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Book chapters on the topic "Number theory – Multiplicative number theory – Primes in progressions"

1

Tao, Terence, and Tamar Ziegler. "Narrow Progressions in the Primes." In Analytic Number Theory. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-22240-0_22.

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2

Friedlander, John B. "Counting Primes in Arithmetic Progressions." In Analytic Number Theory. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-22240-0_7.

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3

Murty, M. Ram. "Primes in Arithmetic Progressions." In Problems in Analytic Number Theory. Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4757-3441-6_12.

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Murty, M. Ram. "Primes in Arithmetic Progressions." In Problems in Analytic Number Theory. Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4757-3441-6_2.

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5

Friedlander, John. "Primes in Arithmetic Progressions and Related Topics." In Analytic Number Theory and Diophantine Problems. Birkhäuser Boston, 1987. http://dx.doi.org/10.1007/978-1-4612-4816-3_7.

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6

Pintz, János. "Patterns of Primes in Arithmetic Progressions." In Number Theory – Diophantine Problems, Uniform Distribution and Applications. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-55357-3_19.

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7

Elliott, P. D. T. A. "Products of Shifted Primes. Multiplicative Analogues of Goldbach’s Problems, II." In Analytic and Elementary Number Theory. Springer US, 1998. http://dx.doi.org/10.1007/978-1-4757-4507-8_12.

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8

Indlekofer, Karl-Heinz, and Nikolai M. Timofeev. "A Mean-Value Theorem for Multiplicative Functions on the Set of Shifted Primes." In Analytic and Elementary Number Theory. Springer US, 1998. http://dx.doi.org/10.1007/978-1-4757-4507-8_9.

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9

"Primes in arithmetic progressions." In Analytic Number Theory. American Mathematical Society, 2004. http://dx.doi.org/10.1090/coll/053/18.

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10

"Primes in Arithmetic Progressions." In Monographs in Number Theory. WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812562272_0009.

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Journal articles
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