Beyond the LSD method for the partial sums of multiplicative functions

Area of Science:

• Number Theory
• Analytic Number Theory

Background:

• The Landau-Selberg-Delange method provides asymptotic formulas for partial sums of multiplicative functions.
• Existing methods often assume strong error terms for prime value averages, leading to precise asymptotic formulas.

Purpose of the Study:

• To investigate the impact of weaker error terms in prime value averages on the asymptotic formulas for partial sums of multiplicative functions.
• To develop novel techniques for estimating partial sums when prime value information is less precise.

Main Methods:

• Adaptation of the Landau-Selberg-Delange method.
• Development of new analytical techniques to handle weaker error bounds.

Main Results:

• Established a framework to derive estimates for partial sums even with limited precision in prime value averages.
• Demonstrated that weaker error terms in prime value averages yield correspondingly weaker, yet still useful, error terms in the partial sum estimates.

Conclusions:

• The study provides a more flexible approach to estimating partial sums of multiplicative functions.
• The developed techniques offer practical implications for number theoretic analyses where precise prime value data is unavailable.