Explicit Bound for the Prime Ideal Theorem in Residue Classes

TL;DR

This paper provides explicit numerical bounds for the generalized Chebyshev functions, aiding in the estimation of computational complexity for algorithms generating special primes used in elliptic curve construction.

Contribution

It introduces explicit bounds for the prime ideal theorem in residue classes, which are crucial for computational number theory applications.

Findings

Explicit bounds for generalized Chebyshev functions provided.

Improved estimates for prime generation algorithms.

Applications to elliptic curve construction using complex multiplication.

Abstract

We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating of computational complexity of algorithms which generates special primes. Such primes are needed to construct an elliptic curve over prime field using complex multiplication method.