On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II

Giovanni Coppola; Saverio Salerno

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 93-102
  • ISSN: 1246-7405

Abstract

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This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials ( a n + b ) ( c n + d ) in short intervals n ∈ [ x , x + x ϑ ] , where now ϑ ∈ ( 0 , 1 ] . Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level ϑ , reaching 3 ϑ / 2 (thus improving also the level of the previous paper, i.e. 3 ϑ - 3 / 2 ), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

How to cite

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Coppola, Giovanni, and Salerno, Saverio. "On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II." Journal de théorie des nombres de Bordeaux 13.1 (2001): 93-102. <http://eudml.org/doc/248705>.

@article{Coppola2001,
abstract = {This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(an + b)(cn + d)$ in short intervals $n \in [x, x + x^\vartheta ]$, where now $\vartheta \in (0,1]$. Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level $\vartheta $, reaching $3\vartheta /2$ (thus improving also the level of the previous paper, i.e. $3\vartheta - 3/2$), but our new results are different in structure. Then, we make a graphical comparison of the two methods.},
author = {Coppola, Giovanni, Salerno, Saverio},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {93-102},
publisher = {Université Bordeaux I},
title = {On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II},
url = {http://eudml.org/doc/248705},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Coppola, Giovanni
AU - Salerno, Saverio
TI - On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 93
EP - 102
AB - This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(an + b)(cn + d)$ in short intervals $n \in [x, x + x^\vartheta ]$, where now $\vartheta \in (0,1]$. Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level $\vartheta $, reaching $3\vartheta /2$ (thus improving also the level of the previous paper, i.e. $3\vartheta - 3/2$), but our new results are different in structure. Then, we make a graphical comparison of the two methods.
LA - eng
UR - http://eudml.org/doc/248705
ER -

References

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  1. [1] E. Bombieri, Le grand crible dans la théorie analytique des nombres. Astérisque18, Société mathématique de France, 1974. Zbl0292.10035MR371840
  2. [2] G. Coppola, S. Salerno, On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals. Functiones et ApproximatioXXVIII (2001), 75-81. Zbl1041.11063MR1823993
  3. [3] H. Halberstam, H.E. Richert, Sieve methods. Academic Press, London, 1974. Zbl0298.10026MR424730
  4. [4] C. Hooley, On the greatest prime factor of a quadratic polynomial. Acta Math.117 (1967), 281-299. Zbl0146.05704MR204383
  5. [5] H. Iwaniec, Almost-primes represented by quadratic polynomials. Inventiones Math.47 (1978), 171-188. Zbl0389.10031MR485740
  6. [6] Ju.V. Linnik, The dispersion method in binary additive problems. American Mathematical Society, 1963. Zbl0112.27402MR168543
  7. [7] S. Salerno, A. Vitolo, On the distribution in the arithmetic progressions of reducible quadratic polynomials. Izwestiya Rossiyskoy AN, Math.Series58 (1994), 211-223. Zbl0839.11040MR1307065

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