Certainly! Let's write a test case generator to create valid inputs for the "Expression Mining" problem. This generator will produce both the integer `n` and the string `s` representing the arithmetic expression. Given the constraints, the generated test cases will ensure that the length of expressions and the depth of nesting are within allowed limits.

Here's the Python code for the test case generator:

```python
import random

def generate_expression(depth=0):
    if depth > 0 and random.choice([True, False]):
        return f"({generate_expression(depth-1)})"
    
    expr_type = random.choice(['single', 'add', 'mult'])
    
    if expr_type == 'single':
        return random.choice('123456789')
    elif expr_type == 'add':
        return generate_expression(depth) + '+' + generate_expression(depth)
    elif expr_type == 'mult':
        return generate_expression(depth) + '*' + generate_expression(depth)
    
def tcgen(num_cases=1):
    cases = []
    total_length = 0
    
    for _ in range(num_cases):
        n = random.randint(1, 10**9)
        max_len = min(2 * 10**6, 5 * 10**6 - total_length)
        length = random.randint(1, max_len)
        depth = random.randint(0, 1000)
        
        s = generate_expression(depth)
        
        while len(s) > length:
            s = generate_expression(depth)
        
        total_length += len(s)
        cases.append((n, s))
        
        if total_length >= 5 * 10**6:
            break
    
    return cases

# Generate test cases
test_cases = tcgen(num_cases=5)

# Print the test cases
for n, s in test_cases:
    print(n)
    print(s)
print(0)  # End of input marker
```

### Explanation:

1. **generate_expression(depth=0)**: This function recursively generates a valid arithmetic expression based on the given grammar. It ensures that the depth of the parentheses does not exceed the specified limit.
   - If the depth is greater than 0, there is a chance to create a nested expression with parentheses.
   - Depending on the randomly chosen expression type, it generates a single digit, an addition expression, or a multiplication expression.

2. **tcgen(num_cases=1)**: This function generates multiple test cases.
   - It initializes the total length of all expressions to ensure it does not exceed the allowed limit.
   - For each test case, it generates a random target value `n` and a random arithmetic expression `s` with a specified maximum length and depth.
   - The expressions are generated until their combined length reaches the maximum allowed total length.

3. **Printing the test cases**: After generating the test cases, it prints them according to the problem's input format. Finally, it prints `0` to mark the end of input.

This generator ensures that all generated expressions are valid and conform to the specified constraints, making it suitable for creating diverse test cases for the "Expression Mining" problem.