Geodesic
A few factors from the Euler product are sufficient for calculating the zeta function with high precision
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 192-202 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The paper demonstrates by numerical examples a nontraditional way to get high precision values of Riemann's zeta function inside the critical strip by using the functional equation and the factors from the Euler product corresponding to a very small number of primes. For example, the three initial primes produce more than 50 correct decimal digits of $\zeta (1/4+10\kern 1pt\mathrm i)$.
Keywords: Riemann's zeta function, functional equation
Mots-clés : Euler product. 
@article{TM_2017_299_a11,
     author = {Yu. V. Matiyasevich},
     title = {A few factors from the {Euler} product are sufficient for calculating the zeta function with high precision},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {192--202},
     year = {2017},
     volume = {299},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_299_a11/}
}
TY  - JOUR
AU  - Yu. V. Matiyasevich
TI  - A few factors from the Euler product are sufficient for calculating the zeta function with high precision
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2017
SP  - 192
EP  - 202
VL  - 299
UR  - http://geodesic.mathdoc.fr/item/TM_2017_299_a11/
LA  - ru
ID  - TM_2017_299_a11
ER  - 
%0 Journal Article
%A Yu. V. Matiyasevich
%T A few factors from the Euler product are sufficient for calculating the zeta function with high precision
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2017
%P 192-202
%V 299
%U http://geodesic.mathdoc.fr/item/TM_2017_299_a11/
%G ru
%F TM_2017_299_a11
Yu. V. Matiyasevich. A few factors from the Euler product are sufficient for calculating the zeta function with high precision. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 192-202. http://geodesic.mathdoc.fr/item/TM_2017_299_a11/

[1] Bui H.M., Gonek S.M., Milinovich M.B., “A hybrid Euler–Hadamard product and moments of $\zeta '(\rho )$”, Forum math., 27:3 (2015), 1799–1828 | DOI | MR | Zbl

[2] Gonek S.M., “Finite Euler products and the Riemann hypothesis”, Trans. Amer. Math. Soc., 364:4 (2012), 2157–2191 | DOI | MR | Zbl

[3] Gonek S.M., Hughes C.P., Keating J.P., “A hybrid Euler–Hadamard product for the Riemann zeta function”, Duke Math. J., 136:3 (2007), 507–549 | MR | Zbl

[4] Gonek S.M., Keating J.P., “Mean values of finite Euler products”, J. London Math. Soc. Ser. 2, 82:3 (2010), 763–786 | DOI | MR | Zbl

[5] Grosswald E., Schnitzer F.J., “A class of modified $\zeta $ and $L$-functions”, Pac. J. Math., 74:2 (1978), 357–364 | DOI | MR

[6] Yu. V. Matiyasevich, “Riemann's zeta function and finite Dirichlet series”, St. Petersburg Math. J., 27 (2016), 985–1002 | DOI | MR | Zbl

[7] Yu. V. Matiyasevich, A few factors from the Euler product are sufficient for calculating the zeta function with high precision, POMI Preprint No 08/2016, St. Petersburg Dep. Steklov Math. Inst., St. Petersburg, 2016

[8] Matiyasevich Yu., Finite Euler products, http://logic.pdmi.ras.ru/ỹumat/personaljournal/eulereverywhere

[9] Pérez Marco R., Statistics on Riemann zeros, E-print, 2011, arXiv: 1112.0346 [math.NT]

[10] Tao T., “A remark on partial sums involving the Möbius function”, Bull. Aust. Math. Soc., 81:2 (2010), 343–349 | DOI | MR | Zbl

[11] S. M. Voronin, “Theorem on the ‘universality’ of the Riemann zeta-function”, Math. USSR, Izv., 9:3 (1975), 443–453 | DOI | MR | Zbl