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A pseudo zeta function and the distribution of primes
1Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA.
Summary
The Riemann zeta function
Area of Science:
- Number Theory
- Analytic Number Theory
- Complex Analysis
Background:
- The Riemann zeta function is central to number theory, with its complex zeros related to prime number distribution.
- The Riemann hypothesis conjectures that all complex zeros lie on the critical line Re(s) = 1/2.
Purpose of the Study:
- To investigate the connection between the complex zeros of the Riemann zeta function and the distribution of prime numbers.
- To explore a modified zeta function based on an idealized prime distribution.
Main Methods:
- Analytic continuation of the Riemann zeta function.
- Definition and analysis of a pseudo zeta function C(s) by replacing primes with n log n in the Euler product.
- Investigation of the zeros of the pseudo zeta function.
Main Results:
- The pseudo zeta function C(s), constructed with an idealized prime distribution (p_n = n log n), has no complex zeros.
- The analytic continuation of C(s) exists for Re(s) > 0, with a singularity at s = 0.
Conclusions:
- The absence of complex zeros in the pseudo zeta function suggests that the complex zeros of the Riemann zeta function are intrinsically linked to the irregularities in the actual distribution of prime numbers.
- This study highlights the importance of prime number distribution for understanding the behavior of the Riemann zeta function and the validity of the Riemann hypothesis.
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