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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):
-
the additive charactere 2πiζ
- 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
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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