Python Code:
import numpy as np
import time
import pyopencl as cl
import math

def sieve_simple(limit):
    """
    Perform a simple sieve of Eratosthenes up to 'limit' using half the memory by considering only odd numbers.
    Returns an array of prime numbers up to 'limit'.
    """
    if limit < 2:
        return np.array([], dtype=np.uint32)
    sieve_size = (limit // 2) + 1  # Only odd numbers
    sieve = np.ones(sieve_size, dtype=np.uint8)
    sieve[0] = 0  # 1 is not prime

    sqrt_limit = int(math.isqrt(limit))
    for num in range(3, sqrt_limit + 1, 2):
        if sieve[num // 2]:
            start_index = (num * num) // 2
            sieve[start_index : sieve_size : num] = 0

    primes = np.nonzero(sieve)[0] * 2 + 1
    # Ensure primes do not exceed the limit
    primes = primes[primes <= limit]
    return primes

def count_primes_with_gpu(n):
    """
    Count the number of primes up to 'n' using the Sieve of Eratosthenes
    with GPU acceleration and advanced optimizations.
    """
    # Step 1: Find small primes up to sqrt(n)
    sqrt_n = int(math.isqrt(n))  # Correct boundary without +1
    small_primes = sieve_simple(sqrt_n)
    print(f"Number of small primes up to sqrt(n)={sqrt_n}: {len(small_primes)}")
    
    # Step 2: Initialize OpenCL
    platforms = cl.get_platforms()
    if not platforms:
        raise RuntimeError("No OpenCL platforms found.")
    
    # Select the first platform with a GPU device
    device = None
    for platform in platforms:
        devices = platform.get_devices(device_type=cl.device_type.GPU)
        if devices:
            device = devices[0]
            break
    
    if device is None:
        raise RuntimeError("No GPU device found on any platform.")
    
    ctx = cl.Context([device])
    # Create a command queue with profiling enabled
    queue = cl.CommandQueue(ctx, properties=cl.command_queue_properties.PROFILING_ENABLE)
    
    # Step 3: Define the OpenCL kernel with optimizations
    kernel_code = """
    __kernel void sieve_of_eratosthenes(__global unsigned char *segment,
                                        __global const unsigned int *small_primes,
                                        const unsigned int num_small_primes,
                                        const unsigned int segment_start,
                                        const unsigned int segment_size,
                                        const unsigned int max_n)
    {
        int idx = get_global_id(0);
        if (idx >= num_small_primes) return;

        unsigned int p = small_primes[idx];

        // Find the first multiple of p within the segment
        unsigned int start = (p * p > segment_start) ? p * p : segment_start + ((p - (segment_start % p)) % p);
        
        // Ensure start is odd
        if (start % 2 == 0)
            start += p;

        // Iterate over multiples of p within the segment
        for(unsigned int multiple = start; multiple < segment_start + segment_size && multiple <= max_n; multiple += p * 2)
        {
            if (multiple < segment_start)
                continue;
            unsigned int index = (multiple - segment_start) / 2;
            if(index >= (segment_size + 1) / 2)
                break;
            unsigned int byte_idx = index / 8;
            unsigned int bit_idx = index % 8;
            // Set the bit to 0 (composite)
            segment[byte_idx] &= ~(1 << bit_idx);
        }
    }
    """

    try:
        program = cl.Program(ctx, kernel_code).build()
    except cl.RuntimeError as e:
        # Retrieve and print the build log directly from the program
        try:
            build_log = program.get_build_info(device, cl.program_build_info.LOG)
        except:
            build_log = "No build log available."
        print("Build log:")
        print(build_log)
        raise
    
    sieve_kernel = program.sieve_of_eratosthenes
    
    # Step 4: Segmented Sieve Parameters
    segment_size = 10_000_000  # 10 million numbers per segment
    remaining_numbers = n - sqrt_n
    num_segments = (remaining_numbers + segment_size - 1) // segment_size
    print(f"Total segments to process: {num_segments}")
    
    # Step 5: Allocate Small Primes Buffer
    small_primes_np = np.array(small_primes, dtype=np.uint32)
    small_primes_buffer = cl.Buffer(ctx, cl.mem_flags.READ_ONLY | cl.mem_flags.COPY_HOST_PTR, hostbuf=small_primes_np)
    
    # Step 6: Initialize Total Prime Count (including prime number 2)
    total_primes = len(small_primes) + 1  # Including prime number 2
    
    # Step 7: Process Each Segment
    for seg_idx in range(num_segments):
        low = sqrt_n + seg_idx * segment_size
        high = min(low + segment_size, n + 1)
        
        # Ensure low is odd and >=3
        if low < 3:
            low = 3
        if low % 2 == 0:
            low += 1
        current_segment_size = high - low
        sieve_bits = (current_segment_size + 1) // 2  # Only odd numbers
        sieve_bytes = (sieve_bits + 7) // 8  # Ceiling division for bytes
        
        # Initialize the segment sieve: all bits set to 1
        sieve = np.ones(sieve_bytes, dtype=np.uint8) * 0xFF
        
        # Clear bit for number 1 if in the first segment
        if seg_idx == 0 and low <= 1 < high:
            index = (1 - low) // 2
            byte_idx = index // 8
            bit_idx = index % 8
            sieve[byte_idx] &= ~(1 << bit_idx)
        
        # Create buffer for the segment
        sieve_buffer = cl.Buffer(ctx, cl.mem_flags.READ_WRITE | cl.mem_flags.COPY_HOST_PTR, hostbuf=sieve)
        
        # Set global work size
        global_work_size = len(small_primes)
        
        # Set kernel arguments with NumPy uint32 types
        sieve_kernel.set_args(
            sieve_buffer,
            small_primes_buffer,
            np.uint32(len(small_primes)),      # arg#3: num_small_primes
            np.uint32(low),                    # arg#4: segment_start
            np.uint32(current_segment_size),   # arg#5: segment_size
            np.uint32(n)                       # arg#6: max_n
        )
        
        # Enqueue the kernel correctly
        try:
            event = cl.enqueue_nd_range_kernel(
                queue,
                sieve_kernel,
                (global_work_size,),
                None  # Let OpenCL decide the local work size
            )
        except cl.RuntimeError as e:
            print(f"Error enqueueing kernel on segment {seg_idx}: {e}")
            build_log = program.get_build_info(device, cl.program_build_info.LOG)
            print("Build log:")
            print(build_log)
            raise
        
        # Read back the sieve data after kernel execution
        sieve_host = np.empty_like(sieve)
        try:
            cl.enqueue_copy(queue, sieve_host, sieve_buffer, is_blocking=True, wait_for=[event])
        except cl.RuntimeError as e:
            print(f"Error copying data from GPU on segment {seg_idx}: {e}")
            raise
        
        # Calculate the maximum number in this segment
        max_num = high - 1  # Since range is [low, high)
        max_num = min(max_num, n)
        
        # Calculate the number of bits to count
        bits_to_count = (max_num - low) // 2 + 1  # Only odd numbers
        
        # Ensure bits_to_count does not exceed sieve_bits
        bits_to_count = min(bits_to_count, sieve_bits)
        
        # Count the number of set bits (primes) in the segment
        bits = np.unpackbits(sieve_host)[:bits_to_count]
        primes_in_segment = np.count_nonzero(bits)
        
        # Add to the total prime count
        total_primes += primes_in_segment
        
        # Optional: Progress update
        progress = (seg_idx + 1) / num_segments * 100
        print(f"Processed segment [{low}, {high}): {primes_in_segment} primes found. Progress: {progress:.2f}%")
    
    return total_primes

# Example Usage
if __name__ == "__main__":
    test_values = [100, 1000, 10000]  # Add more values as needed

    for n in test_values:
        print("\n" + "-"*50)
        print(f"Counting primes up to {n}")
        start_time = time.time()
        prime_count = count_primes_with_gpu(n)
        end_time = time.time()
        print(f"Total number of primes up to {n}: {prime_count}")
        print(f"Execution time: {end_time - start_time:.2f} seconds")

However, as you can see, the code's output: 
--------------------------------------------------
Counting primes up to 100
Number of small primes up to sqrt(n)=10: 3
Total segments to process: 1
Processed segment [11, 101): 25 primes found. Progress: 100.00%
Total number of primes up to 100: 29
Execution time: 0.02 seconds

--------------------------------------------------
Counting primes up to 1000
Number of small primes up to sqrt(n)=31: 10
Total segments to process: 1
Processed segment [31, 1001): 161 primes found. Progress: 100.00%
Total number of primes up to 1000: 172
Execution time: 0.01 seconds

--------------------------------------------------
Counting primes up to 10000
Number of small primes up to sqrt(n)=100: 24
Total segments to process: 1
Processed segment [101, 10001): 1208 primes found. Progress: 100.00%
Total number of primes up to 10000: 1233
Execution time: 0.01 seconds

BUT:
Those are not the actual number of primes.

Please note that I am using a laptop that has an integrated AMD gpu and I am using jupyter notebook.