Complex-Order Riemann Zeta Function Theory and Its Applications to Prime Distribution: From Explicit Zero Equations to the Hardy–Littlewood Conjectures

Key Points

• The aim is to derive explicit equations for the nontrivial zeros of the Riemann zeta function and explore prime distribution.
• Utilized contour integral and hypergeometric representations of the complex-order Riemann zeta function.
• Derived an explicit integral equation for the nontrivial zeros.
• Obtained a refined formula for the prime counting function.
• Extended the theory to the distribution of twin primes with an explicit formula including the Hardy–Littlewood constant.
• Constructed a complete system of explicit formulas for Hardy–Littlewood prime k-tuple conjectures.
• Derived explicit integral equations for nontrivial zeros to enhance understanding of prime distribution.
• Created a refined prime counting function formula that provides better accuracy in counting primes.
• Established an explicit formula for twin primes, incorporating critical constants.
• Developed multiple analytic representations for the Hardy–Littlewood conjectures, enriching theories of prime distribution.