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A new expression for the product of the two Dirichlet series. I. (English) Zbl 1114.11072

Summary: A new expression for the product of the two Dirichlet series is given. From this new expression we derive many results. Among other things we give here another proof of J. R. Wilton’s expression [Proc. Lond. Math. Soc. (2), 31, 11–17 (1930; JFM 56.0289.02)] for the product of two Riemann zeta functions through our new expression. Other results including mean value theorems will be treated elsewhere.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions

Citations:

JFM 56.0289.02
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Full Text: DOI Euclid

References:

[1] Atkinson, F. V.: The mean value of the Riemann zeta-function. Acta Math., 81 , 353-376 (1949). · Zbl 0036.18603 · doi:10.1007/BF02395027
[2] Bateman, H., and Erdélyi, A.: Tables of Integral Transforms. vol. 1. McGraw-Hill, New York (1954). · Zbl 0055.36401
[3] Bateman, H., and Erdélyi, A.: Higher Transcendental Functions. vol. 2. McGraw-Hill, New York (1953). · Zbl 0143.29202
[4] Bellman, R.: An analog of an identity due to Wilton. Duke Math. J., 16 , 539-546 (1949). · Zbl 0035.34002 · doi:10.1215/S0012-7094-49-01649-X
[5] Motohashi, Y.: Spectral Theory of the Riemann Zeta-Function. Cambridge Univ. Press, Cambridge (1997). · Zbl 0878.11001
[6] Titchmarsh, E. C.: The Theory of the Riemann Zeta-Function. Oxford Univ. Press, Oxford-New York (1951). · Zbl 0042.07901
[7] Wilton, J. R.: An approximate functional equation for the product of two \(\zeta\)-functions. Proc. London Math. Soc. (2), 31 , 11-17 (1930).
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