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There exist infinitely many n>0 with p_n^2>p_(n-i)p_(n+i) for all i<n, where p_n is the nth prime. Also, there exist infinitely many n>0 such that 2p_n<p_(n-i)+p_(n+i) for ...

Define E(x;q,a)=psi(x;q,a)-x/(phi(q)), (1) where psi(x;q,a)=sum_(n<=x; n=a (mod q))Lambda(n) (2) (Davenport 1980, p. 121), Lambda(n) is the Mangoldt function, and phi(q) is ...

A generalization of the Gaussian sum. For p and q of opposite parity (i.e., one is even and the other is odd), Schaar's identity states ...

A cyclotomic field Q(zeta) is obtained by adjoining a primitive root of unity zeta, say zeta^n=1, to the rational numbers Q. Since zeta is primitive, zeta^k is also an nth ...

For p an odd prime and a positive integer a which is not a multiple of p, a^((p-1)/2)=(a/p) (mod p), where (a|p) is the Legendre symbol.

The double-struck capital letter I, I, is a symbol sometimes used instead of Z for the ring of integers. In contrast, the lower case symbol i is used to refer to the ...

The double-struck capital letter I, I, is a symbol sometimes used instead of Z for the ring of integers. In contrast, the lower case symbol i is used to refer to the ...

The least positive integer m^* with the property that chi(y)=1 whenever y=1 (mod m^*) and (y,m)=1.

R^- denotes the real negative numbers.

R^+ denotes the real positive numbers.