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VIDEO LIBRARY |
À.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
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Approximation of the zeta function via finite Euler products Yu. V. Matiyasevich St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences |
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Abstract: Consider finite Euler product $$ \zeta_{m}(s)\,=\,\prod_{k=1}^{m}(1-p_k^{-s})^{-1}, $$ where $$ g(s)\,=\,\pi^{-\frac{s}{2}}(s-1)\Gamma\bigl({s}/{2}+1\bigr) $$ is the factor from the functional equation. Modified symmetrized finite xi function $$ \xi^{ :=}_{{m}}(s) \! =\!s^m(1\!-\!s)^m\big(\xi_{m}(s)\!+\!\xi_{m}(1-s)\big) $$ trivially satisfies the functional equation $$ \xi^{ :\text{reg}=}_{{m}}(s)\,=\,\xi^{ :=}_{{m}}(s)-\sum_{k=1}^\infty r_k/(s-q_k). $$ Regularized finite Euler product $$ \zeta^{\approx}_{{m}}(s)\,=\,\xi^{ :\text{reg}=}_{{m}}(s)/\big( s^m(1-s)^m g(s)\big) $$ gives surprisingly good approximations to the values and zeroes of the zeta function. Example 1. The least (in absolute value) non-real zero of function Example 2. The three first Euler factors produce more than 30 correct decimal digits of For more examples visit http://logic.pdmi.ras.ru/~yumat/personaljournal/ eulereverywhere. Language: English |