On the mean values of integral functions represented by Dirichlet series

C ^OMPOSITIO M ^ATHEMATICA

S ATYA N ARAIN S RIVASTAVA On the mean values of integral functions represented by Dirichlet series

Compositio Mathematica, tome 18, n

1-2 (1967), p. 13-16

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13

represented by ^Dirichlet

by

Satya

Consider the Dirichlet series

and

Let _03C3c and _03C3a be the abscissa of convergence and the abscissa of absolute convergence,

respectively,

f(s). If 0’0

infinite,

according

([I],

4)

Dirichlet series which satisfies

(1.1)

equal

f(s) represents

integral

The mean value of

f(s)

We extend this to

where k is any

positive

We shall obtain some of the

properties

m2,k(03C3)

^I2(03C3).

1 This work has been supported by Senior Research Fellowship ^{award of} C.S.I.R., ^{New Delhi} (India).

14

2

THEOREM 1. Let

f(s)

integral function. Then, for

where k is _any

positive

PROOF: From

(1.2)

(1.3),

From

(2.1)

and the

inequalities

I2(x)

increasing

of x.

We may ^note that if

f(s)

integral function,

a constant, ^and

03B1(0

1)

3

THEOREM 2.

Il f(s)

integral function o f

p(O

~), type

type

PROOF: From

(1.3),

where

Taking limits,

get

Also,

(2.1),

since

I2(x)

increasing

Further,

(1.2),

we have

where

p(a)

|aN(03C3)|e03C303BBN(03C3)

N(a),

for Re s ⁼ _a, in the series for

f(s). Therefore,

(3.3)

(3.4),

we

get

Taking limits,

get

or

Since left hand side is

independent

h,

making

we get

The result

(3.1)

easily

(3.2)

(3.5)

functions of finite order

4

THEOREM 3. Let

f(s)

integral function.

16

where

M(03C3)

l.u.b. |f(03C3+it)|

positive

-~t~

PROOF. Since

I2(x) is

increasing

of x,

from

(2.1),

Taking ^limits,

get

Also,

(1.2),

Therefore,

(4.1)

(4.2),

REFERENCE V. BERNSTEIN

[1] Leçons ^{sur les} progrès récents de la théorie des séries de Dirichlet, ^Gauthier Villars, Paris, ^1933.

Department of Mathematics and Astronomy, (Oblatum 4-10-66). ^Lucknow University, ^Lucknow (India).