ON SOME GENERALIZATIONS OF THEOREMS BY LANDAU AND POLYA BY

EMIL GROSSWALD * ABSTRACT

If F(s), the Mellin transform of f(x) (real valued), satisfies certain regularity conditions and if its behaviour on the abscissa of convergence is known, then theorems of Landau and P61ya give information concerning sign changes of f(x). In the present paper, corresponding conclusions are obtained when F(s) does not satisfy the regularity conditions of the theorems of Landau P61ya. 1. Introduction. Let s = ~r + it (0r, treal) be a complex variable and suppose that for a > cro, f ( x ) x -~ ~L(a,b)and that f ( x ) 2 0 for x > a; then it is trivial toobserve that for tr > troand t #O, f~ f ( x ) . ~ dx > [ f~ f ( x ) i r dx [. This simple remark has been made precise and useful by Landau (see [3]), who proves a theorem equivalent to the following statement: TrmOREM A (Landau). Consider the integral S~ f ( x ) x - * d x , where f ( x ) is real and positive for real x > x o. I f the abscissa of convergence is tr = 0 # + oo, then the function F(s), represented for a > 0 by the integral, has a singularity at s = O . FromTheorem A follow almost trivially the following corollaries, which are those actually invoked in most applications. COROLLARY A1. Let f ( x ) be rea Iva lued for rea I x >-_Xo; if F(s) = ~ f ( x ) x -" dx is analytic and regular for r > O, but not regular in any half-plane tr > 0 with ~ >O,and if s = 0 is a point of regularity for F(s), then f(x)changes signs at all points of an infinite set {x.}, with lim.-.ooxn = + oo COROLLARY A2. I f ~ f ( x ) x -~ dx converges for tr>=tro and f(x)>=O for x >=xl (where x~ may be arbitrarily large), then there are only two possibilities for the function F(s) represented (for ~ >=~ro) be the integral: either F(s) is an entire function, or else F(s) is analytic and regular in a half-plane ~ > O, but in no half-plane tr > O - ~ (5 > O) and in that case s = O is a singular point for F(s). Received October 15, 1965 * Written with the support ofthe National ScienceFoundation through the grant GP-3137.

212

E. GROSSWALD

[December

A theorem similar to that of Landau is due to P61ya [5] ; at the price of a slight strengthening of the assumptions, this theorem permits one to obtain a lower bound for the number W(y) of sign changes of f ( x ) in a given interval x o < x < y, y ~ oo. The original proof of P61ya contains a slight gap, pointed out to Ingham by P61ya himself (see [1]).This fact has been mentioned repeatedly in the literature (see e.g. [2]). Actually, the gap can be filled and the theorem, as originally stated by P61ya, is correct. It may be formulated as follows: THEOREM B (P61ya). Let f ( x ) be real for x >=Xo and suppose that the integral Sxof(X)X dx converges in some half-plane tr > tro and that the function F(s), represented (for tr >=tro) by the integral is analytic and regular for tr > O, is not regular in any larger half-plane tr > 0 - e(6 > 0), but is meromorphic in some half-plane ~ >=0 - eo with eo > O. Denote by P the (perhaps empty) set of poles p = tr + it ofF(s) of abscissa a = 0 and define ), = limp ,pinf[ t[ /f P ~- r ~ = + ~ if P = r I f W(y) stands for the number of sign changes of f ( x ) in the interval x o <=x <=y, then limsupy~oo W(y)/logy > y/:r O0

-- 5

Both theorems were proven in view of, and were used for specific applications to number theoretic problems. In his paper, P61ya remarks that these theorems do not seem quite adapted to handle other problems of a similar nature, in analytic number theory. It is the purpose of this paper to strengthen both theorems, by removing some of their more restrictive assumptions, which are often not satisfied in applications (in particular, the requirement of regularity at s -- 0 in Theorem A and that of meromorphisms in Theorem B); this permits their use in many new problems and, in particular, in the proofs of f~-theorems. Two further theorems, which could be used in similar situations are also included in the present paper. Some applications to problems that cannot be handled by the theorems of Landau and of P61ya are appended as illustrations of the method. The four statements obtained are all essentially known, but the present proofs of by now almost classical results seem to be particularly short and neat. Further applications leading to new results are to appear elsewhere. 2. Statement of results. THEOREM 1. Let f ( x ) be real for real x > x o and suppose that f ~ f ( x ) x - ~ d x converges for tr > ao and represents there an analytic function F(s), regular for tr>O, but not for tr > O - e if e > O. Then f ( x ) changes signs at a set of abscissae x = x,, with lim,-.~xn = + ~ , provided that limr247 F(s) = + oo

and that there exists a t #

0 such that either (1) lim sup,-.o+ 1['F(tr +

iO]lF(a) l> 1;

or (2) lim sup,_.o+ [ [F(tr + it)]/F(tr)l = 1 and limr247 {I F(tr + it) [ - I F(a) l} = + ~ "

COROLLARY. Let f ( x ) satisfy the conditions of Theorem 1; then f ( x ) changes

19651

ON THEOREMS BY LANDAU AND P6LYA

213

sign infinitely often, and for arbitrarily large values of x, provided that F(s) satisfies any one of the following conditions: (i) F(s) has a pole at s = 0 + it,limr ) = + oo,lim~_.o+(a-O)F(tr ) = O; (ii) F(s) has a pole at s = O + it (t •O) and a pole of lower order at s=O; (iii) F(s) has poles of the same order k at s = 0 + it and s = 0, with principal parts. k

E

k

a.

and n ~= |

.=t (s-O-it)"

b.

(s-O)"

respectively, with a. + b. (q < n < k), [a,_, [ > I bq_ I (here the integer q satisfies 2 < q < k + I, q = k + 1 meaning la l > Ib l); (iv) F(s) = clog (s - 0) -1 + c'log ( s - O - it)-1 + G(s) with I c' I > I el and G(s) analytic and regular for tr > 0 - e and some e > 0 . THEOREM 2. Let f ( x ) be real .for x > Xo and suppose that the integral f ~ f ( x ) x - S d x converges for a > tro and represents in that half-plane a function F(s), having the following properties: (1) F(s) is regular for tr > O, but not in any half-plane a > 0 - e with e > 0; (2) there exists a denumberable set S ={s. = tr. + it.} without finite limit point satisfying O - e o < a . < 0 for some ~o > O, and such that F(s) can be continued as a meromorphic function in the open set D obtained by making the cuts s = tr + it. (tr < an) in the half-plane a > 0 - eo; (3) for s ~ s . ( s e D ) , F(s) = P . ( s - s . ) . l o g ( 1 / s - s . ) + F.(s), where F.(s) is analytic and regular at s = s. and P.(u) is a polynomial of (effective) degree k , with 0 <=k,, < k - 1, where k is a natural integer depending only on F(s) (i.e., independent of n). Let P = {Sin = an, + itm} be the set of poles of F(s) with 0 - eo < cr,~< 0 and denote by Z the (possibly empty) set {s = 0 + it ls e P L) S} of singularities of F(s) on the line a = O . Set y = l i m i n f ~ z l t [ , if Zv~ ~ ; ),= +oo if Z = ~ . Under these conditions the following holds: I f y ~ O, then f ( x ) changes sign at at an infinite sequence x,, of abscissae with lim._~oox.= + oo; moreover, if W ( y ) = ~,~.l<_y is the number of sign changes for x o < x < y ; then limsupy_.oo W(y)/log y > y/Tt. THEOREM 3. Let f ( x ) be a function of the real variable x, such that: (1) f (x) is real for X >= X o ; (2) jcooox 1-. Idfl converges for a > O where 0 < 1 ; (3) lim~o+ f ~ x t - * [ d f l = + oo; (4)for tr>O, lim.~_.+o~xt-~f(x) = O. Assume also that there exists a t r O, such that

x ) x -s dx lira sup

o

f~

f ( x ) x -~dx

1-0

Itl

then, selecting y arbitrarily large, f ( x ) cannot stay monotonic for x > y.

214

E. GROSSWALD

[December

COROLLARY. I f conditions (1) to (4) hold and f(x) is monotonic for sufficiently

large values of x, then f~ )(x)x-~dx

1- 0

limsup

<

TI~Om~M 4. Let f(x) be a function of a real variable x, satisfying conditions (1) to (4) of Theorem 3 and such that f ( x ) = g ( x ) + h(x), with

f~g(x)x-~ convergent and h(x) monotonic for x sufficiently large (say, for x ~ y ) ; then

lim sup if--cO+

I,I

oO

3. Proof of Theorem 1. Let us assume that condition (1) holds; if the conclusion does not hold, then there exists a y(_-> Xo) such that f(x) keeps a constant sign for x => y and, without loss of generality, we may assume thatf(x) >_-0 for x => y. Let us choose now X ( > y ) arbitrarily large and set G(s)= SXof(X)x-Sdx, an entire function; denote also G(O) = a, G(O + it) = b. Then (1) implies that there exists an eo > 0 and a sequence ~ > 0 , with limn-.ooe, = 0 such that 0 < F ( 0 + ~,) = L~o + J'~' f (x)x-~176 = a + ~, + f ~ f ( x ) ~ - ~ 1 7 6d~ <= (1 - 6)] F(O +8~+ it)l = ( 1 - 6 ) ] b + tl'n+ f~176 I with 6 > 0 independent o f s n ( < Co), and r/n --*0, t/'. -*0, for n ~ oo. The hypothesis liml-.0+F(a) = + ov implies that f~f(x)x -~ dx--* + oo when e. ~ 0 ; hence, by dividing by it, one obtains

f x~(X)X-~

> l+r/,

fx~f(x)x -~ ~"dx with r/> 0, independent of an(< %). Hence

using now the assumption f(x) > Ofor x > X(>_- y), this implies (1 + r/)

f?

f(x)x-~

<

f/

f(x)-~

with ~/> 0,

1965]

ON THEOREMS BY LANDAU AND POLYA

215

which is absurd. This proves Theorem 1 under assumption (1). Assuming (2), the corresponding proof is entirely similar and may be suppressed. REMARK. Comparing this almost trivial proof with the much longer and more difficult one of Landau's Theorem in [3], one may wonder about the reason for this difference. A moment's reflection reveals this as being due to the fact that Landau's Theorem does not assume (as the present one does) that limr F(o) = + oo and still permits one to detect a singularity at s = 0 , even if lim,,_.o.F(o) = O, as may very well be the case.

Proof of the Corollary. In the first, second and fourth case, one verifies that

F(s) satisfies condition (1) of the Theorem; in the third case, F(s) satisfies condition (2).

4. Proof of Theorem 2. Let us assume first that Xo _>-1 and set g(x) = ( - 1)klogkx. f(x); then .[~og(X)x-~dx converges for every tr > % and represents there a function G(s), analytic and regular for o > % and satisfying G(s) = F(k)(s). G(s) is analytic and regular at all points where F(s) is regular and is uniform at least in D; also, at the poles of order r of F(s), G(s) has poles of exact order r + k. Furthermore, for s ~s::~ k

G(s) = F ( % ) =

~

Cam

+ Ffk): " .,

so that the s, are poles of order =< k for G(s). It follows that G(s) is actually uniform not only in D, but at least in the half-plane o > 0 " % ; it is meromorphic there, but not regular in any half-plane a > 0 - t with ~ > 0, because the logarithmic singularities of F(s) are poles of G(s), with nonvanishing principal parts and, denoting the strip { O - e < o < O } by B~, by assumption (1), Be n (P u S) # ~ for every 8 > 0. Consequently, G(s) = .(~g(x)x-Sdx satisfies the conditions of Theorem B and if V(y) stands for the number of sign changes of g(x) in xo < x < y, then limsup V(y) > 7__ y-,~o logy = ~ " However, if W(y) stands for the number of sign changes of f(x) in the same interval, then W(y)= V(y). Finally, we may remove the restriction Xo >= 1, by observing that otherwise V(y) < W(y) < V(y) + 1 and limsup V(y) = limsup W(y) y~oo log'y y~oo logy still holds. 5. Proof of Theorem 3. We observe that iff(x) is monotonic for x ~x~, then

f(x) may change its sign at most once in xt < x < + oo; hence, there exists x2 such that f(x) is monotonic and of constant sign for x ~ x2. From 0 _-<1 and

216

E. G R O S S W A L D

lDecember

(4) it follows that limx_, oof(x) = 0; hence, let us assume, without loss of generality, that for x > xz,f(x) is positive and monotonically decreasing to zero. The function G(s)= f~,~of(X)x-'dx is an entire function and if we set IG(0)l=a, Ic,(o+it)l=b, xt2-~ then O < a , b , c < +oo. For a > O, f,,~of(x)x-~dx = G(s) + f~2f(x)x-'dx, the last improper integral being convergent. Indeed, integration by parts of the Stieltjes integral yields

ff

(*)

f(x)x-'dx

= ~xl-" _sf(X) X

2

1 -l s

f xx X-,df(x)

X2

X2

and, letting X--, oo, the second member becomes 1 1-s

{X 1-7(X) -

x2t-y(x2) -

f; ~ x 1-s-/

~

s

' 1 (r 2- " + :2 -~df(x)).

-

Here the last integral converges by assumption, because a > 0 and the second member of (*) approaches a finite limit; hence, the first member approaches the same limit. Using the fact thatf(x) is positive and monotonically decreasing for x > x2

Id:l

xi-'d: <=

=

1-~

--

X2

so that

~f2

I S~o:~,~-'~ I: I~

~ ~1~"+

< IG(s)l + ~ 1( ~ +

:xT'-,,:~s,,l

fl ~' ~ ,_:1<~:1) 9

For s ~ 0 + + it, this yields:

I oo

f:

~)-'dx

I< b + ~, + ~/Itl + ,r ~ -

where ~I, ~2 ~ 0

::oxl_:ld/i

{

for a ---,0+ . Similarly,

io :~x-~--~ f: 1

1

1 + e2 f f ~ x itI , '-Gdfl ,

~o

= 1 - Z a~

~

x

1 -a

oo

f~,:':'~=~ (

Idf[.

I

(c+

oo

~, f~o~'-'1.) (I: )-'} X 1 -o

1 + ((a - l l G ( a ) -I- c)

2

df

.

1965]

ON THEOREMS BY LANDAU AND POLYA

For a - , O +,

G(tr) ~ a,

217

the bracket becomes 1+83(83 --+ 0 for a - , 0 +,

because

so that

I

< 1-0

1-k~ 2

-

1 §

Itl

1-0

Itl

contrary to the assumption of Theorem 3; consequently, f ( x ) cannot become monotonic for large x and Theorem 3 (and, also its Corollary) are proven. oo

oo -0 dx = A, f~og(x)x dx = B; then [ f ~ f ( x ) x -s dx[ = ] f ~ g ( x ) x - S d x + f~oh(x)x-~dx[ = In +~1 + f~hx)x-'dx[, (st --, 0 for a --, 0 +) and ~Zf(x)x-'dx LTg(x)x-'dx+IZh(x) ax = e + e 2 + f ~ h ( x ) x -~/x = (1 - e3) Ix~oh(x) x - ~ x where e a = (B + e2)(fx~oh(x)x-"dx) - 1---~0

6. Proof of Theorem 4.

Let fxog(x)x

-O-it

=

for tr ~ 0 +. Hence

oo and, dividing numerator and denominator by [ Sxo h(x)x ratio becomes

I I e'~ +

--~

dxl, for a

_.). 0 -I-

this

h(x)x_Sd x o

I (1 + e3) -x X

§

t~5"-~'O"

X

,I x o

The result now follows from the Corollary to Theorem 3. 7. Applications.

Setting, as is usual, A (n)= /l~ (0

if n # pro,

let ~F(x) -- ~E~ x A (n) for x ~ Z, ~e(n) = 89 - 0) + ~F(n + 0)} for n e Z. Then it is known that - ('(s) / s((s) = .f~ ~F(x)x -s- 1 dx, for a > 1. Consequently, if 0 = limsup {tr]((tr + it) = 0} then, for any constant C and 2 < 0 < tr, one has

218

E. GROSSWALD

f

+

[December

)§

-

OU ervin

that F(s) is analytic, regular for a > 0and meromorphic in the whole plane, Theorems A and B are both applicable and one concludes that if C > 0 is arbitrarily large, 1 < x < y and y ~ oo, then ~ ( x ) - x > Cx a and ~ ( x ) - x < - Cx z 9

W(y) >

hold both for at least W(y) distinct values of x, with hmsupy_.OOl-~y =

u

where y > 14 (it is well known that the "smallest" complex root of ((s) is p l = 8 9 + 14.13...0. If, however, ;t = 0, then F(s) has a singularity s = 0 and Theorems A and B do not apply any more. One may, however, use Theorem 1 and its Corollary and obtain THEOREM 5. Let P = { p = O + it[((p)=O}, denote by kp the multiplicity of the zero p of((s), suppose that P # ~j and define m = lim supp~ e kp/[ P I" Then IF(x) - x > Cx ~ and IF(X) - x < - Cx ~ are both satisfied for (naturally different) infinite sequences x = xn, with lim,-.oox, = + oo, provided that (0 <=)C < m. REMARK 1. The condition that P ~ ~ is essential; not only is it needed in order to apply Theorem 1 and its Corollary, but otherwise the statement itself need not hold, as one may verify by considering what happens if 0 = 1, when -

x

=

o(x).

2. The particular case C = 0, 2 < 0 of Theorem 5 is due to P61ya who proved Theorem B specifically for this purpose. The case C > 0, 2 = 0 occurs in E. Schmidt [6], who obtains that without any hypothesis (i.e. even assuming that 0 = 89 the inequalities ~,(x) - x > Cx 1/2 and ~,(x) - x < - Cx 1/2 hold each for an infinite sequence x = x. ~ oo, provided that C = 1/29. The present method yields the slightly better constant ,-e( C = ~

1) ~

14.14

'

but both results fall far short of that by Littlewood (see [4]). 8. S o m e further applications. For x r Z, let f ( x ) = ~,~
~,,
xl12

xl/2

w i t h f ( n ) = ~ { f ( n - O ) + f ( n + 0)} if n e Z; then f ( x ) = it(x) + "-~gx + 0 ,log2--~ .

~" log dyy " It may be shown that Define also, as usual, lix -- .12 ~f(x)

- li x X

x-~dx =

~{ log((s - 1)((s)) + G(s)}

1965]

ON THEOREMS BY LANDAU AND P(SLYA

219

(where G(s), like later also H(s), K(s), L(s), ... are all entire functions without further interest). Consequently, Cx ~

f ( x ) - lix +

_

logx x_,d x = 1 {log((s-1)~(s)) + H(s)} + Clog ~

1

= F(s).

s

Indeed,

oo dx = xs logx

e-(S- 1)y _dy __ K(s) + Y

dx ~ +

0o

e - u dnu.

~-1)

u '

finally,

f ~ 1 7 6 e -" .du ( s -l )

=log

U-

. ff

. du .

s-l) ~

ff

s-l)

1 - e-" du + f f ~e - u -du U

1 ~ (l--s) n (--1) n s " ~ - - n = 1 n ! - - - 7 + ,=1 ~ +

U

e_udu = u log

+ L(s)

and, writing s + 1 - 2 for s the result follows. If :t < 0, then Theorem B is not applicable, but Theorem 2 still is and leads to the following conclusion, which we state as a theorem. THEOREM 6. I f l <--x < y and y ~ oo , then if 2 < 0 , the inequalities f ( x ) - l i x > Cx ~ and f ( x ) - l i x < - C x ~ both hold for at least W ( y ) d i s t i n c t values of x and, regardless how large C has been selected, limsup W(y) > v-oo logy =

__~ (~ = 14.13 ...).

If I = 0, Theorem 2 is no longer applicable but observing that s- 1 log((s - 1)~(s)) = -(kp/p)log((s - p ) - l ) + 0(1) for s --,p, Theorem 1 still yields the following result: THEOREM 7. I f the (-function has zeros p of order kp on the line a = O, let m = lira supp =0+,,kp/lPl;then, it 0 < C < m, the inequalities f ( x ) - l i x > C x ~ x) and f ( x ) - l i x < -Cxe/(logx) are both satisfied, each for an infinite sequence x n of values of x, with x, ~ oo (n ~ 0o). Observing that for 0 > 89 f ( x ) - n ( x ) = o(xe/(logx)), Theorem 7 implies the following TEHOREM 8. I f 0 > 89 and if the zeta-function has zeros of abscissa O, then 7r(x)- lix > Cx~ and ~r(x)- lix < - CxO/(logx) are both satisfied for an

infinite sequence of values x n ~ oo, where C is a (sufficiently small) absolute positive constant.

220

E. GROSSWALD

In case 0 = 89 Littlewood [4] has obtained the results ~g(x)- x = fl+(x 1/2

(xl/21~176 logloglogx) and f ( x ) - lix = f~• ~-logx

x

] , stronger than the present

ones. In order to obtain Littlewood's results, one would have to consider (on the Riemann hypothesis) the function

fl

~176

_

+ Cxt/210gloglogx)x - t - ~ x

X

~s~(s) +

+

loglog ~

+ G(s) = r(s)(G(s)

regular for a > 89 If p = 89+ it o is a zero of ~(s), then, for tr -o 89

F(a + ito) = p(s kv - p) (1 + o(1))and F ( a ) = ~ _ ~ 1 0 g l o g ~ _ l ~ (1 + 0(1)); consequently, for

'

F(a)

Clplloglog a-l~

One then would have to show that for C sufficiently small, this is not possible, but that there exists a C > 0, such that I F ( a + it) ] F(tr) <

kp I I 1 CIpll~176

1 for t r - 8 9 o'- 89

sufficiently small. Our Theorems 1 to 4 are insufficient for that purpose. BIBLIOGRAPHY

1. A. E. Ingham, Aeta Arithmetiea, 1 (1936), 201-211. 2. S. Knapowski and P. Tur,'ln, Aeta Mathematica--Ac. Scient. Hungaricae 13 (1962), 299-314. 3. E. Landau, Mathematisehe Annalen, 61 (1905), 527-550. 4. J. E. Littlewood, Comptes Rendus, Acad. Sciences, Paris, 158 (1914), 1869-1872. 5. G. P61ya, Nachrichten tier kgl. Gesdlschaft der Wissensehaften GOttingen (1930) pp. 19-27. 6. E. Schmidt, Mathematische Annalen 57, (1903), 195-204. UNIVERSITYOF PENNSYLVANIA, PHILADELPHIA