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Two notes on the Riemann Zeta-function*

Published online by Cambridge University Press:  24 October 2008

J. E. Little-wood
J. E. Little-wood
Affiliation:
Trinity College, Cayley Lecturer
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Extract

Let Λ (n) be the arithmetic function usually so denoted, which is zero unless n is a prime power pm (m ≥ 1), when it is log p. We write as usual

and

where the dash denotes that if x is an integer the last term Λ (x) of the sum is to be taken with a factor ½. We wrute further

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 3 , 20 September 1924 , pp. 234 - 242
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

† Landau, E., “Über einige Summen, die von den Nullstellen der Riemannschen Zetafunktion abhängen,” Acta Math., 35 (1911), pp. 271–294.CrossRefGoogle Scholar

‡ Our convention as to the use of O implies, of course, that the formula holds “uniformly for y ≥ √x.”

* I hope to publish this result shortly in another paper.

* Since all roots with γ > 0 lie in 0 < σ < 1, and are symmetrically disposed about the line σ = frac12;

† s 0 is a pure imaginary, and s describes the real axis as ζ describes the large rectangle.

* Here and in the argument below we use the inequality

A glance at the inequalities immediately following verifies the condition |z| < ½.

* Γ4 extends to the left (only) as far as σ = − 19.

† We naturally choose the determination of log ζ defined by the Dirichlet series for σ > 1.