A bound for the least Gaussian prime w with a

Arch. Math. 74 (2000) 423 ± 431 0003-889X/00/060423-09 $ 3.30/0  Birkhäuser Verlag, Basel, 2000

Archiv der Mathematik

A bound for the least Gaussian prime w with a < arg …w† < b By HAJIME MATSUI

Abstract. We give an explicit function B…q† such that there is a Gaussian prime w with ww < B…b ÿ a† and a < arg …w† < b.

1. Introduction. In the present paper, we consider the following problem; for given …a; b† with a < b % a ‡ p2 , estimate the minimum of norms of Gaussian primes whose arguments are within …a; b†. (An element w 2 Z‰iŠ is called a Gaussian prime if …w† ˆ wZ‰iŠ is a prime ideal of Z‰iŠ.) We can give an answer for the problem under ªGRH.º Theorem 1. Assume the truth of the Generalized Riemann Hypothesis for 1 P yr ……a†† L…s; yr † ˆ with yr ……a†† ˆ exp…4 ir arg …a†† and r 2 Z, where a runs over non4 a jaj2s zero elements in Z‰iŠ. Then for any real numbers a; b; with a < b % a ‡ p2 , there exists a Gaussian prime w with a < arg …w† < b such that ww <

A1 …b ÿ a†2

log 4

1 ; bÿa

where A1 is a positive absolute constant. The proof of Theorem 1 employs classical analytic methods for the Hecke L-functions with Grössencharacters, using a special integral kernel in [2]. Moreover, we make use of certain trigonometric polynomials in [4], [5], which are majorants or minorants of the characteristic function of interval …a; b† on the unit circle. Next, we consider whether one can say something without GRH. Theorem 2. For any real numbers a; b; with a < b % a ‡ p2 , there exists a Gaussian prime w with a < arg …w† < b such that ! 3 A2 1 2 ww < exp p log ; bÿa bÿa where A2 is a positive absolute constant. Mathematics Subject Classification (1991): 11R44.

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As a matter of fact, we can get similar results for any imaginary quadratic field (Theorem 3, 4, in the text). 2. Hecke L-functions and a special integral kernel. First we summarize Heckes results which are used in this paper. From now on, weargue pabout not only the Gaussian field but also imaginary quadratic number fields. Let Q ÿd be an imaginary quadratic field, and ÿd its discriminant. Let c be a Grössencharacter of conductor …1† such that  u p a for all a 2 Q ÿd c……a†† ˆ jaj with an integer u. If u ˆ 0, then c is a character of the ideal class group. We define for a complex number with Re…s† > 1 L…s; c† ˆ

X c…I† I

NI s

ˆ

Q P

…1 ÿ c…P†NPÿs †ÿ1 ;

where I runs over all integral ideals, P runs over all prime ideals and NI is the norm of I. We set    n odc  p juj d s G s‡ L…s; c†; L…s; c† ˆ s…1 ÿ s† 2 2p where dc ˆ 1 if c  1, and 0 otherwise. Hecke showed that L…s; c† is analytically continued to an entire function on the whole s-plane, and satisfies the functional equation L…s; c† ˆ iÿjuj W …c†L…1 ÿ s; cÿ1 †; where W …c† is the Gaussian sum of c, and jW …c†j ˆ 1. Our first aim is to compute the following integral in two ways. We set … 1 L0 ÿ …s; c†k…s; x; y†ds Ic ˆ L 2pi …2†

2‡iT …

… with y > x > 1, where

ˆ lim …2†

T!‡ 1

k…s; x; y† ˆ k…s† ˆ



and 2ÿiT sÿ1

y

ÿ xsÿ1 sÿ1

2

;

which is one of the integral kernels used in [2]. The inverse Mellin transform of k…s† is given for v > 0 by 8 0 if v % x2 ; > > > > 1 v > > > log 2 if x2 % v % xy; > … 2pi >1 y2 > > …2† if xy % v % y2 ; log > > > v v > : 0 if y2 % v:

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One method to compute Ic is based on the logarithmic derivative formula of L-function: ÿ

1 PP L0 …s; c† ˆ c…Pn † log NP
NPÿns : L P nˆ1

Thus we have Ic ˆ

1 PP P nˆ1

n b c…Pn † log NP
k…NP ; x; y†;

b Here the sum over those ideals Pn for and this is a finite sum by the above property of k. n which NP is not a rational prime is evaluated in [2]; for y > x ^ 2, P y n b c…Pn † log NP
k…NP ; x; y†  log y
log
…x log x†ÿ1 : x NPn : not prime Hence we have …1†

Ic ˆ

  y b c…P† log NP
k…NP; x; y† ‡ O log y
log
…x log x†ÿ1 : x NP: prime P

Next, using Cauchys theorem, we can show P …2† Ic ˆ dc k…1; x; y† ‡ ÿk…; x; y†; 

where  runs over all zeros of L…s; c†. The case in which c is a character of the ideal class group is shown in [2], and our case is proved by the similar methods. To estimate the righthand side of (2), we first compute the sum over trivial zeros, that is, the zeros where Re…s† ˆ b < 0. Since L…s; c† is an entire function, L…s; c† has simple zeros at the following points: juj ÿ n; n ˆ 0; 1; 2; . . . 2  ˆ ÿ1; ÿ2; ÿ3; . . . ˆÿ

if

uˆ j 0;

if

u ˆ 0:

Hence we obtain X X 4x2bÿ2 k…; x; y† %  xÿ4 : 2 b<0 b<0 …b ÿ 1† We next compute the rest, which is the sum over non-trivial zeros, that is, the zeros where 0 % Re…s† % 1. We need the following lemma, which is proved by the similar method as [3]. Lemma 1. For any T 2 R, we have X X 1 1;  log …d…jTj ‡ juj ‡ 1††; …g ÿ T†2 jgÿTj % 1 jgÿTj>1 where  ˆ b ‡ ig runs over zeros of L…s; c† with 0 % b % 1; g 2 R, subject to each condition. I . Th e c a s e w h e r e w e a s s u m e G R H . The hypothesis asserts that any non-trivial zero  ˆ b ‡ ig of L…s; c† satisfies b ˆ 12, and thereby X X X X 1 1 k…; x; y† % 4xÿ1  xÿ1 1 ‡ xÿ1 : 2 g2 1 j ÿ 1j 0%b%1

bˆ2

jgj % 1

jgj>1

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Hence by Lemma 1 we obtain P k…; x; y†  xÿ1 log …d…juj ‡ 1††: 0%b%1

I I . Th e c a s e w h e r e w e d o n o t a s s u m e G R H . Lemma 2. There exists an absolute constant c > 0 with the following property. In the region defined by c Re…s† ^ 1 ÿ ; jIm…s†j % 1; log …d…juj ‡ 1†† L…s; c† has no zeros with the possible exception of a real simple zero when c  1. P r o o f. If u ˆ j 0, the proof is similar to [2]. Suppose that u ˆ 0. The case that c2 j 1 or c  1 is proved in [2]. Thus we can assume c2  1 and c j 1. In [2], it is shown that L…s; c† has no zeros on the above region with the exception of at most one real simple zero. Under the more general condition c j 1, it is a well-known fact that L…s; c† is essentially the L-function of a holomorphic cusp form with respect to G 0 …d†, and the nonexistence of the zeros outside the region is proved in [1]. This completes the proof. We denote the zero out of the region of Lemma 3 by b0 if it exists; then we have X X 4x2bÿ2 X 1 X X 1 k…; x; y† % %4 ‡4 3‡4 : 2 2 ˆjb g j ÿ 1j j ÿ 1j2 j b0 j b0 ˆ ˆ jgj>1 jgj % 1 0 0%b%1 0%b%1 jÿ1j % p1 jÿ1j>p1 3 3   Because of Lemma 1, the first and second terms are O log …d…juj ‡ 1†† . To estimate the third term we need a lemma, which can be shown with the same method as Lemma 2.2 of [2]. Lemma 3. Let nc …l; s† denote the number of zeros  of L…s; c† with js ÿ j % l. Then for 1 Re…s† ^ 1 and 0 < l % p 3 nc …l; s†  1 ‡ l log …d…jtj ‡ juj ‡ 1††: 1 j b0, it follows from Lemma 2 that j ÿ 1j ^ 1 ÿ b If  satisfies j ÿ 1j % p ;  ˆ 3 c c > . Let s denote ; then by Lemma 3 we obtain log …d…juj ‡ 1†† log …d…juj ‡ 1†† p1 …3 X 1 1 ˆ dnc …l; 1† 2 2 l j ÿ 1j j b0 ˆ s jÿ1j % p1 p1 3  p1 …3 nc …l; 1† 3 nc …l; 1† ‡2 dl ˆ l2 l3 s s

 log 2 …d…juj ‡ 1††: Thus we obtain a final estimate without GRH: P k…; x; y†  log2 …d…juj ‡ 1††: j b0 ˆ 0%b%1

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Summarizing the previous results, we have 8   > < O xÿ1 log …d…juj ‡ 1††   Ic ˆ dc k…1† ‡ O…xÿ4 † ‡ > : O log 2 …d…juj ‡ 1†† ÿ dc k…b0 †

case

I;

case

II:

Since it follows from the Taylor series expansion for k…s† about s ˆ 1 that y k…1; x; y† ˆ log2 , we finally conclude from (1) and the above that for y > x ^ 2 x  b ÿ1  P y 0 ÿ xb0 ÿ1 2 2 y b c…P† log NP
k…NP; x; y† ÿ dc log ‡ dc b0 ÿ 1 x NP: prime ( …3† xÿ1 log …d…juj ‡ 1†† case I; y log y
log
…x log x†ÿ1 ‡ x case II: log2 …d…juj ‡ 1†† 3. Extremal trigonometric polynomials. In this section, we make use of specific extremal trigonometric polynomials, namely Sÿ K …x† below. (Here we say that S…t† is a trigonometric K P polynomial of degree K if S…t† is in the form of ar sin 2prt ‡ br cos 2prt, where ar ; br are rˆ0

real numbers.) Notations are those in [4]. Let t…t† denote the saw-tooth function ( t ÿ ‰tŠ ÿ 12 t 2j Z; t…t† ˆ 0 t 2 Z; where ‰tŠ denotes the largest integer not exceeding a real number t, and     K  1 X r pr 1 1ÿ BK …t† ˆ ÿ cot ‡ sin 2prt K ‡ 1 rˆ1 K‡1 K‡1 p   1 sin p…K ‡ 1†t 2 ; ‡ sin pt 2…K ‡ 1†2 where K denotes a positive integer. Vaaler showed in [5] that BK …t† ^ t…t† for all t and that if T…t† is a trigonometric polynomial of degree % K such that T…t† ^ t…t† for all t, then „1 1 T…t†dt ^ with equality if and only if T…t† ˆ BK …t†. 2…K ‡ 1† 0 Let C…a;b† …t† denote the characteristic function of open interval …a; b† with a < b % a ‡ 1 in R=Z. This satisfies C…a;b† …t† ˆ b ÿ a ‡ t…t ÿ b† ‡ t…a ÿ t† ˆ b ÿ a ÿ t…b ÿ t† ÿ t…t ÿ a† except when t coincides with a or b; in fact the both right-hand sides equal 1=2 at t ˆ a; b. We put S‡ K …t† ˆ b ÿ a ‡ BK …t ÿ b† ‡ BK …a ÿ t†; Sÿ K …t† ˆ b ÿ a ÿ BK …b ÿ t† ÿ BK …t ÿ a†: It is clear that S K …t† is a trigonometric polynomial of degree at most K, that „1  1 ‡ Sÿ SK …t†dt ˆ b ÿ a  . Defining the r-th K …t† % C…a;b† …t† % SK …t† for all t, and that K ‡1 0

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„1  f ÿ2pirt  Fourier coefficients of S dt, one can prove from the above K …t† by SK …r† ˆ SK …t†e 0 properties that 1  …0† ˆ b ÿ a  Sf ; K K‡1   1 1 f  …r† % ‡ min b ÿ a; with S K K‡1 pjrj

…4†

rˆ j 0:

Let l; m be real numbers with l < m % l ‡ 2p, namely 0 < m ÿ l % 2p, and let …w† denote a principal prime ideal; then C…a;b† ^ Sÿ K implies X b log N…w†
k…N…w†; x; y† l<arg c……w††<m

  arg c……w†† b log N…w†
k…N…w†† 2p N…w†: prime  K X  P ÿ …r†
eir arg c……w†† log N…w†
k…N…w†† b Sf ˆ K X

^

Sÿ K

rˆÿK

N…w†: prime

ˆ

K X rˆÿK

Since

S K …x†

ÿ …r† Sf K

n

P N…w†: prime

o b cr ……w†† log N…w†
k…N…w†† :

ÿ …ÿr† ˆ S ÿ …r†. Thus we have f is real valued, we have Sf K K X b log N…w†
k…N…w†; x; y† l<arg c……w††<m ÿ …0† ^ Sf K

…5†

ÿ2

X

b log N…w†
k…N…w††

N…w†: prime

K X rˆ1

ÿ …r†j jSf K

P N…w†: prime

b : cr ……w†† log N…w†
k…N…w††

This inequality with (4) is a variation of the Erdös-TuÂran inequality ([4], [5]). An upper bound of the left-hand side can be similarly obtained by using S‡ K , but it is no use for our purpose. 4. Proofs of theorems. p Theorem 3. Let F be an imaginary quadratic field Q ÿd with discriminant ÿd, and let w; h be the number of roots of unity in F and the class number. Assume the truth of the Generalized Riemann Hypothesis for L…s; j yr † where j …j ˆ 1; 2; . . . ; h† are all characters of the ideal class group of F, y……a†† ˆ exp…iw arg …a†† for all a 2 F  , and r 2 Z. Then for any 2p real numbers a; b; with a < b % a ‡ , there exists a prime element w in F with w a < arg …w† < b such that N…w† <

C3 log 2 d 2

…b ÿ a†

log4

1 ; …b ÿ a†

where C3 is a positive absolute constant. P r o o f. From now on, c1 ; c2 ; . . . denote absolute positive constants. Since we have u ˆ wr for yr and w % 6 for all imaginary quadratic fields, it follows from (3) that for y > x ^ 2

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y b j yr …P† log NP
k…NP; x; y† ÿ dj yr log 2 x NP: prime o n y % c1 log y
log
…x log x†ÿ1 ‡ xÿ1 log …d…jrj ‡ 1†† : x X

Using this formula and the orthogonality relation of j, we have X y b yr ……w†† log N…w†
k…N…w†; x; y† ÿ hÿ1 dyr log 2 x N…w†: prime o n y % c1 log y
log
…x log x†ÿ1 ‡ xÿ1 log …d…jrj ‡ 1†† : x 3 ÿ Since we have ÿK % r % K, we obtain the inequality Sf …r† with r ˆ j 0 by (4). It is % K 2jrj trivial that wa < arg y……w†† < wb holds if and only if a < arg …w† < b. Now applying (5) for l ˆ wa and m ˆ wb, we obtain X b log N…w†
k…N…w†; x; y† a<arg …w†


 n o w 1 y y …b ÿ a† ÿ log 2 ÿ c2 log y
log
…x log x†ÿ1 ‡ xÿ1 log d 2p h…K ‡ 1† x x K n o X 1 y ÿ3
c1 log y
log
…x log x†ÿ1 ‡ xÿ1 log …d…r ‡ 1†† r x rˆ1   w 1 y ^ …b ÿ a† ÿ log 2 2p h…K ‡ 1† x n o y ÿ1 ÿ c3 log y
log
…x log x† ‡ xÿ1 log …d…K ‡ 1†† log …K ‡ 1†: x ^

We set y ˆ ex; then the right-hand side becomes ^

w 1 …b ÿ a† ÿ ÿ c4 xÿ1 log …d…K ‡ 1††
log …K ‡ 1†: 2p h…K ‡ 1†

We set x ˆ x0 log d with x0 ^ 1; then the right-hand side becomes ^

w 1 …b ÿ a† ÿ ÿ c5 x0ÿ1 log 2 …K ‡ 1†: 2p h…K ‡ 1†

1 < 1=x0. Moreover it follows from the assumption h…K ‡ 1† x0 ^ 1 that log2 …K ‡ 1† % log 2 …2x0 †. Hence we have X b log N…w†
k…N…w†; x0 log d; ex0 log d† We set K ˆ ‰x0 Š; then we have

a<arg …w†
^

w w …b ÿ a† ÿ x0ÿ1 ÿ c5 x0ÿ1 log2 2x0 ^ …b ÿ a ÿ c6 x0ÿ1 log2 2x0 †: 2p 2p

C 1 log 2 and C sufficiently large. If we put q ˆ b ÿ a, then we can bÿa …b ÿ a† make the following satisfying for all q;

We set x0 ˆ

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 1 1 2 c6 q log 2C ‡ log ‡ 2 log log q q > 0: qÿ 1 Clog2 q Thus we can conclude that there exists a prime element w with a < arg …w† < b such that e2 C2 log 2 d 1 log 4 . This implies Theorem 3. N…w† < 2 …b ÿ a† …b ÿ a† Theorem 4. Let F, w and h be as Theorem 3. Then for any real numbers a; b; with 2p a < b % a ‡ , there exists a prime element w in F with a < arg …w† < b such that w ! 3 A4 1 N…w† < exp p log 2 ; bÿa bÿa where A4 is a positive constant depending only on F. P r o o f. From now on, a1 ; a2 ; . . . denote positive constants depending only on F. We put y y ˆ x2 ; then we have log y
log
…x log x†ÿ1  1. Thus it follows from (3) that x P r b y ……w†† log N…w†
k…N…w†† ÿ hÿ1 dyr log 2 x % a1 log 2 …jrj ‡ e†: N…w†: prime Similarly as the proof of Theorem 3, we obtain P b log N…w†
k…N…w†; x; x2 † a<arg …w†
 ^

 w 1 …b ÿ a† ÿ log 2 x ÿ a2 log 3 …K ‡ 1†: 2p h…K ‡ 1†

p 1 We set x ˆ exp f…K ‡ 1†log3 …K ‡ 1†g; then we obtain log2 x % log3 …K ‡ 1†. Hence h…K ‡ 1† we have n o P w b log N…w†
k…N…w†; x; x2 † ^ log3 …K ‡ 1† …b ÿ a†…K ‡ 1† ÿ a3 : 2p a<arg …w†
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[4] H. MONTGOMERY, Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conf. Ser. in Math. 84, Amer. Math. Soc. 1994. [5] J. D. VAALER, Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc. 12, 183 ± 216 (1985). Eingegangen am 11. 1. 1999 Anschrift des Autors: Hajime Matsui Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya 464-8602, Japan