One kind hybrid character sums and their upper bound estimates

Area of Science:

• Number Theory
• Algebraic Number Theory
• Combinatorics

Background:

• Character sums are fundamental in analytic number theory, with applications in primality testing and cryptography.
• Estimating character sums is crucial for understanding their distribution and properties.
• Lucas polynomials and Gauss sums are established tools in number theory with rich properties.

Purpose of the Study:

• To investigate estimation problems concerning hybrid character sums.
• To apply analysis methods, Lucas polynomials, and Gauss sums to derive new bounds.
• To explore applications of these estimates in proving combinatorial identities.

Main Methods:

• Utilizing properties of Lucas polynomials.
• Applying techniques involving Gauss sums.
• Employing analytical methods for sum estimation.

Main Results:

• Several sharp upper bound estimates for hybrid character sums were obtained.
• The derived estimates provide precise control over the magnitude of these sums.
• New combinatorial identities were proven as direct applications.

Conclusions:

• The study successfully provides tight bounds for hybrid character sums.
• The methods employed offer a powerful framework for analyzing such sums.
• The connection to combinatorial identities highlights the broader applicability of the results.