Example functions.php file. This removes the <title> tag and replaces it with the word "Hello!" - Don't run it on a live site!

functions.php

<?php

// This file is loaded as well as the parent's (twentythirteen) child theme, so name your functions with a prefix. Like below:-
function twentythirteen_child_title( $title, $sep ) {
	return "Hello!";
}

// This function is hooked in after the theme is set up. It removes the default title and replaces it with the word "Hello!". Don't run on a live site!
function twentythirteen_child_change_title() {
	remove_filter( 'wp_title', 'twentythirteen_wp_title' );
	add_filter( 'wp_title', 'twentythirteen_child_title', 10, 2 );
}
add_action( 'after_setup_theme', 'twentythirteen_child_change_title' );
?>