iwhich are the real parts of the Dirichlet polynomials а7 ехр(2тиу; г), where ccj =

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIX (1990)

J.

and W.

(Poznan)

On zeros and sign changes of certain Dirichlet polynomials Dedicated to Professor W lady slaw Orlicz on the occasion of his 85-th birth anniversary.

1. In the analytic theory of numbers, the so-called explicit formulae play a very remarkable and interesting role. They express values of arithmetic functions in terms of zeros and poles of zeta-functions (see [3], [11]). A number of such formulae can be written in the form

F (x )= X aQxe + R{x, T), M^r

the summation being over all zeros or poles Q = p + iy of certain zeta- functions with llmgl = |y| ^ T; R (x, T) denotes a “small” remainder. If we write

0 T = sup Reg, x = e\

\y\*ZT

then the formula above can be written as

F* (x) = e~eTtF{x) = X aQeiVi + R*{x, T).

\у\аТ,р = вт

Hence, to some extent, the discussion on the properties of F* (oscillations, changes of sign, and so on) reduces to the investigation of the Dirichlet polynomial

(see f8! f9! C10!)-

The aim of this note is to examine some properties of polynomials of this type.

2. Let уj , ..., yN be given different real numbers and at , . . . , aN, , . . . , bN any real numbers. We shall study functions of the form

(2.1) f(t) = X

cos (2

iyj t + bj)

j =

i

which are the real parts of the Dirichlet polynomials а7 ехр(2тиу; г), where

ccj =

exp (ibj).

We denote the field, and also the topological additive group of real numbers, by the same letter R, and its discrete subgroup consisting of rational integers by Z.

Let M c R denote the Z-module generated by exponents у1?..., yN. M is a free module since it is finitely generated and torsionfree. Hence, there exist numbers /il5 ..., ц„ such that

M = ц х Г® ...® nnZ.

We can suppose that are all positive. Moreover, n ^ N and П

У] = Z Rjm m = 1

with some integers Rjm. Let us denote n-dimensional torus by Q = T 1 x ... x T \

where Г 1 denotes the interval [0, 1) supplied with quotient group R/Z topology. Obviously, the group operation in T 1 is addition mod 1.

Besides, Q has the natural structure of an n-dimensional differentiable manifold.

We define the functions Fk: Q -» R, к = 1, 2 ,..., by the formulae FA*!>•••» Q = Z (Re Z aj (2™Уj)1 ~1 exp(2

f RJmtm))2-

1=1 j= 1 m= 1

Denoting by  the continuous homomorphism Л: R-+Q, Щ = d is til, \\ii2t \\,..., \\fiHt\\)t where ||a|| is the fractional part of a real a, we have

k~ 4 d l У

ы т ) = 1 { < й ' т ) -

Let us write

M* = Ft (t) = 0}

and let Q0 be a subgroup of Q defined by

Q0 = {t = (t1, ..., tn): tj = 0}.

Let n: Q -* Q 0 be the projection

7Г ( t i , . . . , f n)

tj~ —

Ri j=i

Furthermore, let vk: Q0 -* N и {0} be a function defined by

Vjk(t) = # ( я _1(0 } )п М к)

Zeros and sign changes of Dirichlet polynomials 225

( # A denotes the cardinality of A). Later we shall prove that the definition of vk is correct, i.e. the set n ~ 1{{t})r\M k is finite for each к and every te£?0.

Let Nk(T) denote the number of zeros of order к of the function / i n the interval [0, T], T > 0, where each such zero is counted only once.

Now, we are able to state the first result of the present note.

Th e o r e m

1. I f 2, then

' N k(T)

(2.2) lim —— = xk,

Г -00 1 where ,

(2.3) xk = j {vk (t) - v k + 1 (t)) dp (t)

П о

and dp denotes the normalized Haar measure on the group Q0.

The condition n ^ 2 is natural since for n = 1 the function is., periodic and the investigation of the number of its zeros of a given order is trivial in this case.

Since sign changes appear at zeros of odd order, we have the following

Co r o l l a r y.

where

lim

Г - o o

V(T) T = X,

x = J Z h ( t ) - V i ( t ) ) d y i ( t ) .

П о 2 Xk

It can easily be seen that the set M k is empty for sufficiently large k. Thus, the sum £ 2ik(vk(*) — vk+i(*)) is correctly defined.

The problem of the number of zeros of Dirichlet polynomials was the subject of interest of several authors. The existence of the limit (2.2) can be deduced from the results of a joint paper of B. lessen and H. Tornehave (see [6]) but without (2.3), M. Kac proved in [7] that if all exponents are linearly independent over Q and h, = 0, у = 1 ,..., N, then for the number of zeros of the polynomial (2.1) in the interval [ — T, T], we have

— 00 — 00

where / 0 denote a Bessel function of index zero. P. Turân gives in [13] an

upper estimate for the number of zeros of (2.1) in the interval [ a ,a + d],

depending on a, d, N and т а х |у 7|, 1 <y С ЛГ (if a} 0 ,j = 1, ..., N).

Before stating the second theorem, let us introduce some definitions.

Suppose that t1 < t 2 < t3 are consecutive zeros of the Dirichlet polynomial f We call a zero t2 ô-small if

max |/(r)| < 3 or max |/(t)| < <5.

t i ^ t ^ t 2 t 2 * Zt <t 3

Denote by N(T, 3) the number of <5-small zeros of / in the interval [0, T], where each zero is counted according to its multiplicity.

We shall prove that almost all zeros of / are not <5-small.

T

2. For every e > 0, there exists a 3 > 0 such that lim sup

7-00

N(T,Ô)

T

.

I f t x and t2 are some consecutive zeros of the polynomial f and t0e (t1, t2) is such that

|/( t0)| = max |/(t)| > ô,

à < J \f(t)\d t < t2- t x.

ti

We shall call a zero tx o f f 3-isolated if the interval — <5, + <5) is free of zeros different from t t . Denoting by (T, 3) the number of zeros of / in the interval [0, T] which are not ^-isolated, we get from Theorem 2 the following corollary.

C

. For every e > 0, there exists a 3 > 0 such that lim sup

T - o o

N i( T ,8 )

T

.

It means that the zeros of / which are situated “close to each other” are rather exceptional.

3. We say that a mapping is smooth if it belongs to the class C00. Now let us prove that the mapping n is smooth on the set Q \Q 0.

Let px: Q T 1 be a function defined by Pi (t) = Pi f r i , . . . , О = *i,

and let r: T 1 -* R be a function defined by r(t) = t. Obviously, p t is a smooth mapping and r is also smooth except for the point t = 0 where it is not even continuous.

For every t = (tl5 ..., t„), let us determine the value of

a(t) = ido-A (t),

Zeros and sign changes of Dirichlet polynomials 221

where “ —” denotes subtraction in the group Q0. We get

a(t) = t —Я( ^ -rp ^ t)^ = h

h — -» j

Mi

= rc(t).

i=i

Thus we have n = \iQ — X{rp1/p,x). This means that n is a composition of a number of mappings. They are smooth, except for the mapping r at the point

= 0. Therefore, n is smooth on Q \Q 0.

4. Next, let us examine in detail the set M k for a given k. Since the function Fk is analytic on Q, therefore the set M k is analytic and compact in Q.

Hence

M* = M£ u M l, M l n M l = 0 , where

(1) M k is an open subset of M k (in the topology induced on Mk);

(2) M k is an (n — l)-dimensional submanifold of Q;

(3) M l is closed in M k and its dimension is not greater than n — 2.

The existence of such a decomposition of a complex analytic set is a well-known fact from the theory of analytic functions of several variables (see [5]). In the real case, a similar result can be proved as follows. Suppose that U is an open subset of Rn and suppose further that Ü is an open subset of C"

such that Ü n R n = U. We can assume that Ü is symmetrical, i.e. together with the point (zl5 ..., zn) it contains (z1?..., z„). Let / be a real-valued function defined on U, which can be continued as an analytic function over Ü. This analytic continuation will also be denoted by f Thus we have

(4.1) f{ z x, ..., zn) = f( z lt ..., z„).

Let us write

M = {(zl5 ..., z„)eL: f{ z t , ..., z„) = 0}.

Since M is a complex analytic set, there exists a decomposition M = M° u M 1 satisfying conditions (1), (2), and (3). Obviously, the dimensions of the sets mentioned in the formulation of those conditions are understood as the dimensions over the field of complex numbers and, therefore, they are equal to twice the real dimensions.

We shall prove that (M° n Rn) и (M1 n JR") is the needed decomposition of the real analytic set M n R n. Condition (1) is obviously satisfied. To prove condition (2), consider a point p0 e M° n Rn. Let V be a symmetric neighbour­

hood of p0 in C", such that (after appropriate renumbering of coordinates, if necessary)

V n M ° = {(ф(г2, ..., z„), z2, ..., z„): (z2, ..., z j e l '}

for an open set Vt c cn_1 and holomorphic ф. Since, owing to (4.1), the set

V n M ° is symmetric, we have

(ф(г2, ..., z„), z2, zn) e V n M °

for any (z2, ..., z„)e Vt . Hence, ф(z2, ..., zn) = ф(z2, ..., z„) and so ф is real for real (z2, ..., zn). Therefore,

V n (M° n R”) = {(ф (x2, ..., x„), x 2, ..., x„): (x2, ..., x„)e Vt n Rn}

and this proves that M° n R " is a submanifold of U, having the dimension equal to n — 1.

We shall now prove (3). Owing to the Cardan theorem (see [5]), the set M 1 is analytic and its dimension is not greater than n — 2. Hence, if p e M 1 n Rn and the set M 1 is determined in the neighbourhood of the point p, by the formulae

Z l ~ Ф \ ( Z k +

1’ • • •

)>

...

Z k = Ф к ( г к + 1 ’ • • • ’ Z n) ’

where ф(, i = 1, 2 ,. . . , k, are holomorphic and к ^ 2, then the set M 1 n Rn is contained in the submanifold which has dimension equal to n — k, and is determined by the equations

*k = Rеф к(х* + 1, ..., x„),

and (хк+1, . . . , х и)еК 1п Я п"1 Hence, the set M 1 n R" is at most (n — 2)-dimen- sional, and the existence of the decomposition of the set Mk is completely proved.

Since the mapping л:|я^ 0 is smooth and Mk is a submanifold of Q, the mapping

|

/2\

0 ls a^so smo°th. Let К be the set of critical points of this mapping. From the well-known theorem of Sard (see [4]) we have that p (n (К)) = 0. Let us consider

S0 = n (K ) u (Mk n f20) u n{Mk).

Obviously,

p (S0) < p ( n (K)) + p(M k n Q0) + p(n(Ml)),

and since the sets M k n Q 0, n(M k) have dimensions not greater than n — 2 — dim C2

— 1, we have p(S0) — 0.

If tE n (M k) \S 0 then t is a regular value of the mapping п\п^По. Hence,

there exists a neighbourhood U c: n(M k) \S 0 of t which is properly covered by

л. It follows in particular that the set Tt(Mk) \S 0 is open in Q0:

Zeros and sign changes of Dirichlet polynomials 229

5. We shall now prove that the functions vfc are bounded. Let te n (M k).

There exists t0, 0 < t0 < 1//^, such that

+ = a > 0.

Then the function of a real variable

^0, — ^ э и i-> Fk (t + Я (и))

cannot be identically equal to zero in any nonempty interval, because it is the restriction to the interval (0, 1/fij) of a certain function ipk t which is holomorphic in the domain s = <7 +

, |<r| < 2/fix, |r| < 1. Therefore, t0 always exists.

There exists also a neighbourhood U of t such that for every t' eU we have

(5.1) Fk(t + X(t0) ) > U .

Since the functions i/ik>t-, VeU , are uniformly bounded in the domain s = er + rr, \o\ < 2//л1, |

| < 1, hence owing to (5.1), also the numbers of zeros of the functions <pM, are uniformly bounded (from Jensen’s formula, see [12]). It follows that the numbers vk(t'), f e U, are bounded by a constant C(U) depending on U but independent of the choice of f. In other words, the function vfc is locally bounded on n(M k). Since vk = 0 outside n(M k), thus it is locally bounded on Q0. But Q0 is compact, so vk is globally bounded.

From the above, we can easily deduce that the set n(M k) \S 0 can be written in the form

7Г (Mk)\S 0 = U

U2

. . .

Ul , where

Uj = {ten {M k) \S 0: vk(t) = ;}.

The function vk is constant in a neighbourhood of every te n (M k) \S 0 and so Uj are open subsets of n(M k). This follows from the fact that each point te7r(Mk)\iS0 is properly covered by n. Hence the sets n(M k)\(U l u . . . u U ! u Uj+ j u . . . u UL) are closed and therefore

Üj с тг(М*)\(171 и . . . и u UJ+1 u . . . u Uj)

for every j, and so, for the boundary bd Uj of the set U we have bd Uj = Üj \ Uj CZ 7i (Mk) \( f /1 u ... u Uj) = S0.

Hence,

H ( h d

Uj) ^ fi(S0) = 0.

This gives an essential information about the sets Uj since by the classic

8 — Roczniki PTM — Prace Matematyczne XXIX

Kronecker-Weyl theorem (see [1], [2]) on Diophantine approximation we have for every /, 1 < j ^ L,

where

ï l l - # , ) , T - + c o ,

1 k e D j T

(5.2) D;(D = jfc к < T, ( ^ft

f t e t/,

z = i

6. Let iV*(r) be the number of zeros of order not lower than к of the function / in the interval [0, Г].

If t is such that

к k + 1

— ^ t < --- ,

H i Hi

then

7 Г (Я (Г ))

Hit , \\Hit\\Hi Y ( ^кн,

Hi l r ( ^Hi

Consequently,

N * (T )= I v, mui

Hi + 0 (1 )= X ; £ 1+ X 1 + 0(1),

Z = l / j = 1 meD j([/t 1 Г]) meDo

where the sets Dp j = 1 , 2 , L, are defined by (5.2) and 0 0 = D0(T) = <r: r ^ [/^ T], rHi

/'i e 5 f

z = 1

Hence,

N * (T )= i y > 1 T#<(t7j) + 0 (T) = /ii r j vt (t) ^ (t) + o(T).

j = 1 По

But

W* (T) = JVÎ (T) - N U , (T) ~ д, T f (vt (t) - v„ +, (t)) dpi (t) По

and the proof of Theorem 1 is completed.

7. We shall now prove Theorem 2. All the notation introduced previously are still valid. Since we are now interested in zeros of any multiplicity, we shall only deal with the set M t . For instance, the sets U j,j = l , 2 , . . . , L , and S0 are now subsets of я (M J.

Let te и ...

UL. There exists an open neighbourhood V of t

such that F c UL, which is properly covered by the mapping n\Ml.

Zeros and sign changes of Dirichlet polynomials 231

We can also suppose that the closures of the connected components of 7i_1( F ) n M 1 are disjoint. Hence, there exists a positive constant c = c(V), depending on V, such that for every t'e V and every

t' + Afrj), t, + A(t2)G7i“ 1({t'})nM 1, 0 < t x < t2 ^ 1/jUi we have

(7.1) max F x (t' + A(t)) ^ c,

(7.1) max F 1(t' + À (t))^ c,

(7.1) max F 1(t' -\-X(t))^ c,

OsStSïti

Thus, if F c= t/j и ... и UL is a closed subset, then there exists a constant C = C(F)> 0 depending on F and satisfying (7.1), (7.2), (7.3) for every

t' + Aftj), t' + À(t2)e n ~ i ({t'})n M l ,

0 ^ tt < t2 ^ 1/ni, t 'e i 7. This obviously follows from the fact that F is compact.

It follows also from the above that for every open neighbourhood V of S0 there exists a positive number <5 = 0(V) such that, if t0 is a <5-small zero of the polynomial / then n (A (t0)) e V. Since ц (S0) = 0, therefore for every e > 0 one can choose a neighbourhood V such that

H(V) < - 6 , , M(bd V) = 0, 2ц1 m1 к

where

m1 = max v1 (t)

tefio

and к denotes the maximal multiplicity of zeros of /.

Thus we have

where

Finally,

N ( T , S ) ^ k £ v,

me D( T)

m/i, n

Pi i=i ^{+ 0 (1),}

Щ Т) = r: 1 < r ^ O i Г], m Pi

N(T, <5) ^ km1fi{V)fil T+ o(T) < eT,

for T ^ T0 (e) and the proof of Theorem 2 is completed.

References

[1] J. W. S. C a s s e ls , An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge 1957.

[2] K. C h a n d r a s e k h a r a n, Introduction to Analytic Number Theory, Springer-Verlag, Berlin 1968.

[3] H. D a v e n p o r t , Multiplicative Number Theory, Markham Publishing Company, Chicago 1967.

[4] J. D ie u d o n n é , Éléments d ’Analyse, vol. Ill, Gauthier-Villars Editeur, Paris-Bruxel­

les-Montréal 1974.

[5] M. H e r v é , Several Complex Variables, Local Theory, Oxford University Press, Bombay 1963.

[6] B. J e s s e n , H. T o r n e h a v e , Mean motions and zeros of almost periodic functions, Acta Mathematica 72 (1945), 137-279.

[7] M. K ac, On the distribution o f values of trigonometric sums with linearly independent frequences, Amer. J. Math. 65 (1943), 609-615.

[8] J. K a c z o r o w s k i, On sign-changes in the remainder-term of the prime number formula, II, III, Acta Arithmetica 45 (1985), 65-74; ibidem 48 (1987), 347-371.

[9] J. K a c z o r o w s k i, J. P in t z, Oscillatory properties o f arithmetical functions, I, II, Acta Math.

Hungarica 48 (1986), 173-185; ibidem 49 (1987), 441-453.

[10] J. K a c z o r o w s k i, W. S ta s, On sign-changes in the remainder-term o f the prime-ideal theorem, Colloq. Math. 56 (1988), 185-197.

[11] K. P r a c h a r , Primzahlverteilung, Springer-Verlag, Berlin-Gôttingen-Heidelberg 1957.

[12] E. C. T itc h m a r s h , The Theory o f Functions, Oxford University Press, Oxford 1947.

[13] P. T u r â n , Eine neue Methode in der Analysis und deren Anwendungen, Budapest 1953.

(INSTYTUT MATEMATYKI, UNIWERSYTET IM. A. MICKIEWICZA, POZNAN) INSTITUTE O F MATHEMATICS, A. MICKIEWICZ UNIVERSITY

POZNAN, POLAND.