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

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.

Advisor:

School:The University of Hong Kong

School Location:China - Hong Kong SAR

Source Type:Master's Thesis

Keywords:exponential sums divisor theory

ISBN:

Date of Publication:01/01/2003