Some results on the error terms in certain exponential sums involving the divisor function

by Wong, Chi-Yan

(Uncorrected OCR) Abstract of thesis entitled

SOME RESULTS ON THE ERROR TERMS IN CERTAIN EXPONENTIAL SUMS INVOLVING THE DIVISOR FUNCTION

submitted by

Chi-Yan WONG

for the Degree of Master of Philosophy

at The University of Hong Kong

in August 2002

In 1985, Jutila investigated a generalization to the Dirichlet divisor problem. He defined, for a and b being coprime integers and a 2: 1,

6 (x; ~) = ~I d(n)e (b:) - ~ (lOg :2 + 21- 1) - E (0; ~) ,

n

where ~I indicates that the last term in the sum is to be halved if x is an integer, d(n) is the divisor function and E (0; ~) is the value at 8 = 0 of the analytic continuation of the Dirichlet series

(Re (8) > 1).

This thesis is devoted to investigation of certain properties of the error term 6 (x; ~).

The third power moment of the functi,on Re (e-iIl6 (x; ~)) was studied. By following the main idea of Tsang (1992) and an estimation of Lau (1999), the following theorem was proved:

Theorem 1 for 1 ::; a ::; X, and any real e,

where C2 is the constant

00 00

L L I-i(h)h-t (aJ3(a + J3)r~ d(a2h)d(J32h)d((a + 13)210)

( bh ) (bh ) (bh )

. cas 27r~a2 + () cas 27r~J32 + e cas 27r~(a + 13)2 + e .

Here b is the positive integer (mod a) such that bb _ 1 (mod a).

Theorem 1 also implies that

The higher power moments of \.6. (x;~) I were investigated. The following theorem was proved:

1

Theorem 2 For a ::; X"2 ,

jX It. (x; ~)f dx

This gives a non-trivial bound for the integral It 1.6. (x; ~) IA dx for A < 8 under certain restrictions on a. The case A = 3 of Theorem 2 also suggests that the above consequence of Theorem 1 is not the best possible in the sense that the restriction on a is less strict. The proof of Theorem 2 employs an estimate on the number of large values of 1.6. (x; ~) I?

The sizes of the gaps between signichanges of the function Re (e-il1.6. (x; ~)) was studied. It was shown that such gaps can be as large as of order aX~ log-5 X. A mean square estimate of .6. (x + U; ~) - .6. (x; ~) was proved and was then used in the proof of the above result.