# Primes in arithmetic progressions to large Moduli. II

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## Abbreviations

p :

a prime number

Λ(n):

the von Mangoldt function

τ j (m):

the divisor function

ϕ(q):

the Euler function

μ(m):

the Möbius function

e(ζ):

χ(n):

a multiplicative character

$$\hat f$$ :

the Fourier transform off, i.e.,

$$\hat f(\eta ) = \int\limits_{ - \infty }^\infty {f(\xi )e(\xi \eta )d\xi }$$
m≡a(q) :

meansm≡a (modq)

$$\frac{{\bar d}}{c}$$ :

meansa/c (mod 1) wheread≡1 (modc). Sums involving this symbol are restricted, often without explicit mention, to values of the variable for which the function summed is defined

m∼M :

meansM≦m<2M

∥α∥:

meansL 2 norm of α=(α m ), i.e., ∥α∥=(∑|α m |2)1/2

x :

a large number

ℒ:

logx

π(x; q, a):

the number of primesp≦x, p≡a(modq)

Ψ(x; q, a):

$$\sum\limits_{n \leqq x,n \equiv a(\bmod q)} {\Lambda (n)}$$

$$\sum\limits_{b(q)} {^* }$$ :

means the summation over residue classesb(modq) with (b, q)=1

S(a, b; c):

means the Kloosterman sum$$\sum\limits_{m(c)} {^* } e((am + b\bar m)/c)$$

A :

arbitrary large, positive constant, not necessarily the same in each occurrence

B :

some positive constant, not necessarily the same in each occurrence

ε:

any sufficiently small, positive constant, not necessarily the same in each occurrence

## References

1. Bombieri, E.: On the large sieve. Mathematika12, 201–225 (1965)

2. Bombieri, E., Friedlander, J., Iwaniec, H.: Primes in arithmetic progressions to large moduli. Acta Math.156, 203–251 (1986)

3. Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math.70, 219–288 (1982)

4. Dress, F., Iwaniec, H., Tenenbaum, G.: Sur une somme liée à la fonction de Möbius. J. Reine Angew. Math.340, 53–58 (1983)

5. Elliott, P.D.T.A., Halberstam, H.: A conjecture in prime number theory. Symp. Math.4, 59–72 (1968–69)

6. Fouvry, E.: Répartition des suites dans les progressions arithmétiques. Acta Arith.41, 359–382 (1982)

7. Fouvry, E.: Autour du théorème de Bombieri-Vinogradov. Acta Math.152, 219–244 (1984)

8. Fouvry, E.: Théorème de Brun-Titchmarsh; application au théorème de Fermat. Invent. Math.79, 383–407 (1985)

9. Fouvry, E., Iwaniec, H.: On a theorem of Bombieri-Vinogradov type. Mathematika27, 135–172 (1980)

10. Fouvry, E., Iwaniec, H.: Primes in arithmetic progressions. Acta Arith.42, 197–218 (1983)

11. Friedlander, J., Iwaniec, H.: On Bombieri's asymptotic sieve. Ann. Scuola Norm. Super., IV. Ser.5, 719–756 (1978)

12. Friedlander, J., Iwaniec, H.: Incomplete Kloosterman sums and a divisor problem. Ann. Math.121, 319–350 (1985)

13. Heath-Brown, D.R.: Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math.34, 1365–1377 (1982)

14. Iwaniec, H.: Rosser's sieve. Acta Arith.36, 171–202 (1980)

15. Iwaniec, H.: A new form of the error term in the linear sieve. Acta Arith.37, 307–320 (1980)

16. Mardzanisvili, K.K.: The estimation of a certain arithmetic sum (in Russian). Dokl. Akad. Nauk. SSSR22, 391–393 (1939)

17. Shiu, P.: A Brun-Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math.313, 161–170 (1980)

18. Vaughan, R.C.: Mean value theorems in prime number theory. J. Lond. Math. Soc.10, 153–162 (1975)

19. Vinogradov, A.I.: On the density hypothesis for DirichletL-functions. Izv. Akad. Nauk. SSSR Ser. Mat.29, 903–934 (1965); correction ibid. Vinogradov, A.I.: On the density hypothesis for DirichletL-functions. Izv. Akad. Nauk. SSSR Ser. Mat.30, 719–720 (1966)

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Supported in part by NSERC grant A5123

Supported by NSF grant MCS-8108814 (A02)

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Bombieri, E., Friedlander, J.B. & Iwaniec, H. Primes in arithmetic progressions to large Moduli. II. Math. Ann. 277, 361–393 (1987). https://doi.org/10.1007/BF01458321

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• DOI: https://doi.org/10.1007/BF01458321