×

Linearized product of two Riemann zeta functions. (English) Zbl 1310.11082

Summary: In this paper, we elucidate the well-known Wilton’s formula for the product of two Riemann zeta functions. A proof of Wilton’s expression for product of two zeta functions was given by M. Nakajima in [ibid. 79, No. 2, 19–22 (2003; Zbl 1114.11072)] using the Atkinson dissection. On the similar line we derive Wilton’s formula using the Riesz sum of the order \(\kappa =1\).

MSC:

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

Citations:

Zbl 1114.11072
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] F. V. Atkinson, The mean-value of the Riemann zeta function, Acta Math. 81 (1949), 353-376. · Zbl 0036.18603 · doi:10.1007/BF02395027
[2] R. Bellman, An analog of an identity due to Wilton, Duke Math. J. 16 (1949), 539-545. · Zbl 0035.34002 · doi:10.1215/S0012-7094-49-01649-X
[3] K. Chandrasekharan and S. Minakshisundaram, Typical means , Oxford Univ., Press, 1952. · Zbl 0047.29901
[4] G. H. Hardy and M. Riesz, The general theory of Dirichlet’s series , Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964.
[5] M. Nakajima, A new expression for the product of the two Dirichlet series. I, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 2, 19-22. · Zbl 1114.11072 · doi:10.3792/pjaa.79.19
[6] J. R. Wilton, An Approximate Functional Equation for the Product of Two \(\zeta\)-Functions, Proc. London Math. Soc. S2-31 (1930), no. 1, 11-17.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.