Riemann zeta fractional derivative—functional equation and link with primes
Abstract This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-07-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-019-2202-5 |
| _version_ | 1828880291616260096 |
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| author | Emanuel Guariglia |
| author_facet | Emanuel Guariglia |
| author_sort | Emanuel Guariglia |
| collection | DOAJ |
| description | Abstract This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane. |
| first_indexed | 2024-12-13T09:44:54Z |
| format | Article |
| id | doaj.art-1c4863ae164f4d15a3196be34a3e5d28 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-13T09:44:54Z |
| publishDate | 2019-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-1c4863ae164f4d15a3196be34a3e5d282022-12-21T23:52:06ZengSpringerOpenAdvances in Difference Equations1687-18472019-07-012019111510.1186/s13662-019-2202-5Riemann zeta fractional derivative—functional equation and link with primesEmanuel Guariglia0Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico IIAbstract This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane.http://link.springer.com/article/10.1186/s13662-019-2202-5Fractional derivativeRiemann ζ functionFunctional equationCritical stripPrime numbers |
| spellingShingle | Emanuel Guariglia Riemann zeta fractional derivative—functional equation and link with primes Advances in Difference Equations Fractional derivative Riemann ζ function Functional equation Critical strip Prime numbers |
| title | Riemann zeta fractional derivative—functional equation and link with primes |
| title_full | Riemann zeta fractional derivative—functional equation and link with primes |
| title_fullStr | Riemann zeta fractional derivative—functional equation and link with primes |
| title_full_unstemmed | Riemann zeta fractional derivative—functional equation and link with primes |
| title_short | Riemann zeta fractional derivative—functional equation and link with primes |
| title_sort | riemann zeta fractional derivative functional equation and link with primes |
| topic | Fractional derivative Riemann ζ function Functional equation Critical strip Prime numbers |
| url | http://link.springer.com/article/10.1186/s13662-019-2202-5 |
| work_keys_str_mv | AT emanuelguariglia riemannzetafractionalderivativefunctionalequationandlinkwithprimes |