Parity of the number of primes in a given interval and algorithms of the sublinear summation

TL;DR

This paper introduces a new approach to determine the parity of the number of primes in an interval, improving upon existing algorithms by developing sublinear methods and general theorems for summation of multiplicative functions.

Contribution

It presents a novel algorithmic approach that implicitly determines the constant c in the sublinear time complexity for prime parity calculations.

Findings

Developed sublinear algorithms for summation of multiplicative functions

Proved several general theorems related to these summations

Achieved an implicit value of the constant c in the complexity bound

Abstract

Recently Tao, Croot and Helfgott invented an algorithm to determine the parity of the number of primes in a given interval in O(x^{1/2-c+\eps}) steps for some absolute constant c. We propose a slightly different approach, which leads to the implicit value of c. To achieve this aim we discuss the summation of multiplicative functions, developing sublinear algorithms and proving several general theorems.