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Primitive character sums andDirichlet L-functionsTom M. Apostol aa California Institute of Technology, Departmentof Mathematics , Pasadena, California, 91125,USAPublished online: 10 May 2007.

To cite this article: Tom M. Apostol (1978) Primitive character sums and DirichletL-functions , Applicable Analysis: An International Journal, 8:2, 115-123, DOI:10.1080/00036817808839220

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4 p p l m b I e Anaivis, 1978, Vol. 8, pp. 115-123 0 Gordon and Breach Science Publishers Ltd.. 1978 Printed in Great Br~tain

Primitive Character Sums and Dirichlet L - Functionst TOM M . APOSTOL

California Institute of Technology, Department of Mathematics, Pasadena, California 91 725, U.S.A.

(Receiced September 16 , 1977)

Communicated bv David L. Colton

1. INTRODUCTION

Let x be a primitive Dirichlet character modulo k and let G(x) denote the ' Gauss sum

By using two different representations of the Dirichlet L-function L(s, jl) with s = O we derived the identity [I, p. 2261

each member being equal to - kG(f)L(O,x). Subsequently, Berndt [4] gave an elementary proof of (I), that is, a proof that did not use the theory of L-functions.

The functional equation for the L-functions [2, p. 2631 implies

so (1) implies the two equivalent formulas

+Dedicated to Arthur Erdelyi on the occasion of his seventieth birthday

115

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116 T. M. APOSTOL

and

In this paper we give an elementary proof of (2) directly from first principles using no property of L(1,x) except its definition as a series,

the convergence of this series being part of the proof. We also give another elementary proof of (1) which, together with (2), provides an elementary proof of (3). When ~ ( n ) = (n 1 p), the quadratic character modulo an odd prime p -= 3 (mod 4), (3) reduces to the classic formula

The sum on the left is related to a theorem of Dirichlet [ 5 ] which states that for an odd prime p = 3 (mod 4) there are more quadratic residues than nonresidues in the interval (O,p/2). In fact, an elementary argument 161 shows that

if p = 3 (mod 4) and hence

Dirichlet's theorem states that the sum on the left is always positive. The results of this paper show that there is an elementary proof of

identity (4) using no complex function theory, no Fourier series, and no special properties of L(1, x). Thus, an elementary proof of the inequality

would result in an elementary proof of Dirichlet's theorem. The usual proof of (5) invokes the Euler product representation of the L-function L(s)=E,"=, (n 1 p)n-"or real s > 1 and the continuity of L(s) at s = 1. (See Whiteman, [65].)

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PRIMITIVE CHARACTER SUMS AND DIRICHLET L-FUNCTIONS 117

2. BASIC L E M M A S

Throughout this paper we write e(x)=e2"'" for real x and

for nonintegral x. Since

for nonintegral x, and since 1;:: j (h ) =0, the right member of (1) can be written as follows:

where

To prove (2) we need to show that

Our first lemma deals with sums of the type (7 ) in which the numerator %(h) is replaced by a more general arithmetical function.

LEMMA 1 Let f be any arithmetical function, periodic modulo k, such that f ( k ) = 0. Define

F(k ) ="il f(h)cp(h/k). h= 1

Then for ez;ery integer n z 1 we have the multiplicative property

Proof We start with Lagrange's interpolation formula for the constant polynomial 1 and write

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118 T . M . APOSTOL

where A ( x ) = ( x - x,) . . . ( x - xn _ ,) and x,, . . . , xn - , are n distinct complex numbers [3, p. 5801. Take x,=E' where E is any primitive nth root of unity. Then A ( x ) = xn - 1, A 1 ( x r ) = nx:-l= nc- ' and ( 9 ) implies

Now take ~ = e - ~ " ' ~ " and x =e(h /nk ) where h and k are positive integers with 1 5 h < k. Then (10) becomes

Multiply each member of (11) by f ( h ) and sum on h to obtain

Since f is periodic mod k we have f (h )= f (rk+ h). Also as r and h run through the numbers h = 1,2,. . . , k - 1 and r = 0,1 , . . . , n - 1, respectively, the linear combination rk + h runs through those integers 1,2, . . . , nk - 1 which are not divisible by k. Since f vanishes at the multiples of k we can write

This proves Lemma 1. Our strategy now is to exploit the use of the parameter n in the identity

Since m/nk+O as n+ co this suggests approximating its Taylor polynomial. If 0 < 1 t / 5 TC we have

the factor rp(m/nk) by

so when t = (2nimlnk) we get

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PRIMITIVE CHARACTER SUMS AND DIRICHLET L-FUNCTIONS 119

In our application f will be a nonprincipal Dirichlet character mod k so the sum C::,l f ( m ) is zero and I f ( m ) I = 1. This gives us, after summing on m,

Unfortunately the error term O(k) does not tend to 0 as n- x. Although this last result is compatible with (8 ) it does not imply ( a ) , so a modified approach is needed.

The presence of the term x ( - 1 ) in (8) suggests we use the property X ( - l ) x (h ) = X ( - h) = ~ ( k - h ) which, in turn, suggests combining terms symmetrically located in the sum (7). This device, together with the following lemma, will lead to an elementary proof of (8).

LEMMA 2 Let cp(x)= ( e ( x ) - 1 ) - l . Then if 0 < a < b s i we have

b -a d b ) = d a ) + o ( ~ ) .

Proof We use the relation

To estimate the last term we use the Taylor polynomial approximation

valid for / t I 5 n. Taking t = 2nix where 0 <a 5 x 5 b 23 we find

1 - 1 2 -- I

e x ) ( 2 7 - c ~ ) ~

so (13) shows that

in the given interval. This implies (12).

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120 T. M. APOSTOL

3. THE PRINCIPA'L THEOREM

THEOREM 1 Let x be any nonprincipal Dirichlet character mod k and let

For integer n 2 2 let A= [ J n l . Then we have

Note Letting n-cc in (15) we find that the series x,"=, ~ ( h ) / h converges and we obtain the identity

which reduces to (8) when x is replaced by 2.

Proof We write

.- . The last term is 0 if k is o d d and is i X ( [ k / 2 ] ) if k is even. The factor i occurs because q($) = - i. Since ~ ( k - h ) = x ( - l ) ~ ( h ) we have

By Lemma 1 we can also write

where

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PRIMITIVE CHARACTER SUMS AND DIRICHLET L-FUNCTIONS 121

We split the sum into two parts,

Since 3. = [ J n ] and n z 2 we have i s n / 2 hence 3 . k 5 [ n k / 2 ] . To estimate the second sum we further subdivide it into subintervals of length k . Write h = Q k + r , where O s r s k - 1 . Then

But x ( Q k + r ) = ~ ( r ) so

Applying Lemma 2 with

we can replace q ( Q / n + r l n k ) by q ( Q / n ) and introduce an errok

in the curly bracket. Similarly, by applying Lemma 2 to the complex conjugate of cp we can replace

and introduce an error of the same order. This gives us

The expression in curly brackets is now independent of r and C::; ~ ( r ) = 0 so ( 1 9 ) becomes

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122 T. M. APOSTOL

Next we treat the first sum in (18),

The Taylor polynomial approximation in (14) implies

and

Multiplying by ~ ( h ) / n and summing on h we find that (21) implies

I k - 1

h = 1

Each 0-term is O(n-1/2) so from (22), (20) and (17) we obtain (15). Note The constant implied by the 0-symbol in (22) and in (15) is

independent of n but may depend on k. However, this constant is S A k for some absolute constant A.

4. ELEMENTARY PROOF OF (1) USING FINITE FOURIER EXPANSIONS

A short elementary proof of the identity

follows easily from two well-known finite Fourier expansions. For integer r, 1 5 r k - 1, we have (see [2], p. 175)

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PRIMITIVE CHARACTER SUMS AND DIRICHLET L-FUNCTIONS 123

and for any primitive Dirichlet character mod k, we have

(see [2], Theorem 8.20). If we multiply each member of (23) by k G ( f ) ~ ( r ) and sum on r, noting that CFL: x(r)=O, then interchange the order of summation and use (24), we immediately obtain (1).

References

[I] Tom M. Apostol, Dirichlet L-functions and character power sums, Journal of Number Theory 2 (1970), 223-234.

[2] Tom M. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976.

[3] Tom M. Apostol, Calculus, Vol. 11, 2nd Edition, John Wiley and Sons, Inc., New York, 1969.

[4] Bruce C. Berndt, An elementary proof of some character sum identities of Apostol, Glasgow Mathematical Journal 14 (1973), 5&53.

[5] P. G. Lejeune Dirichlet, Recherches sur diverses applications de l'analyse infinitesimale a 18 theorie des nombres, Journal fur die reine und angewandte Mathematik 19 (1839), 324-369.

[6] Albert Leon Whiteman, Theorems on quadratic residues, Mathematics Magazine 23 (1949/50), 71-74.

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