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A similar modification can be made in Theorem 6. If H(o)<π/2v for a>n, then the series is summable (A, v) for

σχη.

For

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for sufficiently large values of x. The remainder of the argument follows the course of the proof of Theorem 6. Combining these results, we see that the region of summability (A, v) is the half-plane in which

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§ 13. Similar methods may be applied to the general series

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We may say that the series Ea, is summable (4, 1) if Eae-y'n is convergent for all positive values of y and tends to a limit when y0. Thus summability (A) is the same as summaybility (A, n), and summability (A, v) as summability (A, n').

If we suppose that Zae-s has a region of absolute convergence, we can prove a series of results analogous to those proved in the preceding sections for ordinary Dirichlet's series. The region of summability (A, 1) is the half-plane in which

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tends to a limit, we may say that Za, is summable (A, l, v). The region of summability (Ă, l, v) of the series Σae-λns is the half-plane in which

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NOTE ON PERRON'S INTEGRAL AND SUMMABILITY-ABSCISSAE OF DIRICHLET'S

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and let f(s) denote the function defined by means of, or associated with, this series (1); in this sense we write

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By "Perron's integral" we mean the integral of the form

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where c, and x are certain real constants, the path of integration being the line σ = c.

As is well known, Perron established the following important theorem*:

If the Dirichlet's series (1) is convergent for s=B, and c> 0, c> ß, then

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*Perron, Journal für Math., vol. cxxxiv. (1908), pp. 95-143. Landau: Handbuch der Lehre von der Verteilung der Primzahlen, pp. 820 et seq. Hardy and Riesz: The General Theory of Dirichlet's Series (Cambridge Tracts in Mathematics and Mathematical Physics, No. 18), p. 12.

By this notation we mean

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m

Here, in the case wλ, the integral is to be regarded as being defined by its principal value.

Landau gives, in his Handbuch,* a corresponding theorem in the case c<0, viz.

If the Dirichlet's series (1) is convergent for s=B, and B<c<0, then

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G. H. Hardy and M. Riesz give, in their book The General Theory of Dirichlet's Series, the following generalisation of the above theorem of Perron:

If the Dirichlet's series (1) is summable (λ, x),§ where K> 0, for 8B, and c> 0, c> ß, then

(4)

where (5)

=

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The formula (3) may easily be deduced from (2) by means of Cauchy's theorem. The principal aim of Section I. of this paper is to obtain a corresponding formula analogous to (4). This forinula cannot be deduced from (4) so simply as (3) can from (2), the difficulty lying in the fact that si is not one-valued when is not an integer.

Let R (w) denote the " sum of the series

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Loc. cit., p. 831.

The same remark (for w=Am) as in the last theorem is also to be applied here. It should be observed that, if A, <w <λm, then 'a, −ƒ(0) = 82. ay

Loc. cit, p. 50.

รม+1

v=m+1

For the meaning of this notation, see Hardy and Riesz, The General Theory of Dirichlet's Series, pp. 21 et seq.

Observe that R°(w) Σ a,=

av (\m<w<\m+1}; the expression

λιλω v=m+1

wR(w) may be considered as a sort of generalised form of remainder of the

series

The principal theorem which I am going to prove is
THEOREM A. If the Dirichlet's series

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is summable (, ), where > 0, for s=ß, ß being negative, and c <0, c> ß, then

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In Section II. I shall give some theorems for means of the second kind, which are analogous to those in Section I.

*

2. Let o denote the abscissa of summability (λ, x) of the Dirichlet's series (1).

The value (7) of Perron's integral with c <0 is interesting, because it occurs in the formula for σ, when σ <0, in a manner analogous to that in which the value (4) of Perron's integral with c>0 occurs when σ>0. I have not yet succeeded in proving this result in full generality, though it seems very probable. I have proved it only in the case in which is integral. As is well known,† the summability-abscissa σ, if positive, is given by

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log | A* (w) |

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In Section III. I shall prove that the summability-abscissa Ok, if negative, is given by

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K being a positive integer.

Formula for the abscissa of convergence of the Dirichlet's series (1), applicable alike whether the abscissa is positive or negative, have been given by T. Kojima,‡ M. Fujiwara,§ and E. Lindh. I have succeeded in generalising Fujiwara's result. As the formulæ are of an entirely different character from those which I have mentioned in the above, I content myself for the present with stating the result only.

For the meaning of this nomenclature, see Hardy and Riesz: The General Theory of Dirichlet's Series, pp. 21 et seq.

† Ibid., p. 45, Theorem 31.

Tohoku Mathematical Journal, vol. vi. (1914), pp. 131–139.

§ Ibid., pp. 140-142.

| G. Mittag-Leffler: Comptes Rendus, 22 Feb. 1915.

The summability-abscissa σ (>0) is given by

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where σ may take any value whatever (positive, negative, or

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where (x) is a real function, satisfying certain conditions.*

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While I have been writing this paper, Mr. G. H. Hardy has kindly given me valuable advice on several points. I take this opportunity to tender him my heartiest thanks.

1. Perron's Integral.

3. Before going into the proof of Theorem A, I will give some lemmas.

LEMMA I. If u is real, and c <0, к> 0, then we have

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u, c and κ being independent of s.†

This lemma may be deduced without difficulty, by means of Cauchy's Theorem, from Hankel's expressiont of the reciprocal of the Gamma function as a contour integral, which is as follows:

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where the integral is taken along a contour line C commencing at positive infinity, running along the upper vicinity of the positive part of the real axis (ie. o-axis) without crossing it,

* If denote the positive integer such that k-1<kk, then the conditions are as follows:

(i) p(x), p′(x), ...,

(ii) lim '(x) = ∞,

(+1)(x) are finite for all finite values of x greater than x,
φ) (2)
(iii) lim
xP

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where p is a certain positive constant.

ρ

=0 (r = 1, 2,...,k+1),

Hardy and Riesz: The General Theory of Dirichlet's Series, p. 50, Lemma 10. Zeitschrift für Mathematik, Jahrgang 9 (1864), pp. 1-21. Or see Whittaker and Watson: Modern Analysis, p. 239.

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