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;

vii