Notation
Particular notation used is collected at the start of the index; some general
notation is described here.
o N, Z, Z+, Q, R, C denote the natural numbers, integers, non-negative
integers, rational numbers, real numbers, and complex numbers, respec-
tively;
o Qp, Zp, Cp denote the p-adic rationals, the p-adic integers, and the com-
pletion of the algebraic closure of Qp, respectively;
o ordp z is the p-adic order of z £ Cp;
o P is the set of prime numbers;
o 1Z is a commutative ring with 1;
o ¥q is a field with q =
pr
elements, p £ P, r N, and F* is its multiplicative
group;
o Fp, p e P is identified with the set {0,1,... ,p 1};
o given a field F, F denotes the algebraic closure of F; thus Q is the field of
all algebraic numbers;
o for any ring K, Tl[X], H(X), K[[X]], K((X)) denote the ring of polyno-
mials, the field of rational functions, the ring of formal power series, and
the field of formal Laurent series over 7£, respectively;
o ZK denotes the ring of integers of the algebraic number field K;
o H(f) denotes the naive height of / G Z[#i,... ,x
m
], that is, the greatest
absolute value of its coefficients;
o gcd(ai,... , a*;) and lcm(ai,... , a^) respectively denote the greatest com-
mon divisor and the least common multiple of a i , . . . , a& (which may be
integers, ideals, polynomials, and so forth);
o e denotes any fixed positive number (for example, the implied constants
in the symbol O may depend on e);
o Sij denotes Kronecker's 5-function: Sij = 1 if i j , and Sij = 0 otherwise;
o /x(fc), (f(k), r(fc), a(k) respectively denote the Mobius function, the Euler
function, the number of integer positive divisors of fc, and the sum of the
integer positive divisors of fc, where k is some non-zero integer;
o i/(fc), P(k), Q(k) respectively are the number of distinct prime divisors of
fc, the greatest prime divisor of fc, and the product of the prime divisors
of fc; thus, for example: i/(12) = 2, P(12) = 3, and Q(12) = 6;
o for a rational r = k/£ with gcd(k,£) 1, P(r) = max{P(/c), P{£)} and
Q(r) = m&x{Q(k),Q(£)};
o TT(X) is the number of prime numbers not exceeding x\
o \X\ denotes the cardinality of the set X;
o log x = log2 x, In x = loge x;
o Logx = logx if x 2, and Logx = 1 otherwise;
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