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- Title
A Dirichlet series related to the error term in the Prime Number Theorem.
- Authors
Elma, Ertan
- Abstract
For a natural number n , let Z 1 (n) : = ∑ ρ n ρ ρ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character χ modulo q ≥ 3 , we define Z 1 (s , χ) : = ∑ n = 1 ∞ χ (n) Z 1 (n) n s for ℜ (s) > 2 and obtain the meromorphic continuation of the function Z 1 (s , χ) to the region ℜ (s) > 1 2 . Our main result indicates that the poles of Z 1 (s , χ) in the region 1 2 < ℜ (s) < 1 , if they exist, are related to the zeros of many Dirichlet L -functions in the same region.
- Subjects
PRIME number theorem; DIRICHLET series; ZETA functions; NATURAL numbers; RIEMANN hypothesis; MEROMORPHIC functions
- Publication
International Journal of Number Theory, 2024, Vol 20, Issue 3, p715
- ISSN
1793-0421
- Publication type
Article
- DOI
10.1142/S1793042124500362