An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution
Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its valu...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2020/1832982 |
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| author | Michael Milgram |
| author_facet | Michael Milgram |
| author_sort | Michael Milgram |
| collection | DOAJ |
| description | Two identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+it, everywhere inside the asymptotic t⟶∞ critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures. |
| format | Article |
| id | doaj-art-77997d880ddf4d378c930698be7f196f |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-77997d880ddf4d378c930698be7f196f2025-08-20T02:07:10ZengWileyAbstract and Applied Analysis1085-33751687-04092020-01-01202010.1155/2020/18329821832982An Integral Equation for Riemann’s Zeta Function and Its Approximate SolutionMichael Milgram0Geometrics Unlimited, Ltd., Box 1484, Deep River, Ontario, K0J 1P0, CanadaTwo identities extracted from the literature are coupled to obtain an integral equation for Riemann’s ξs function and thus ζs indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates ζs anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, both an analytic expression for ζσ+it, everywhere inside the asymptotic t⟶∞ critical strip, as well as an approximate solution can be obtained, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of ζσ+it for different values of σ and equal values of t; this is illustrated in a number of figures.http://dx.doi.org/10.1155/2020/1832982 |
| spellingShingle | Michael Milgram An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution Abstract and Applied Analysis |
| title | An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution |
| title_full | An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution |
| title_fullStr | An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution |
| title_full_unstemmed | An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution |
| title_short | An Integral Equation for Riemann’s Zeta Function and Its Approximate Solution |
| title_sort | integral equation for riemann s zeta function and its approximate solution |
| url | http://dx.doi.org/10.1155/2020/1832982 |
| work_keys_str_mv | AT michaelmilgram anintegralequationforriemannszetafunctionanditsapproximatesolution AT michaelmilgram integralequationforriemannszetafunctionanditsapproximatesolution |