Is it possible to generate an exact 15 kHz clock pulse using an Arduino?
You can get pretty close if you program Timer 1 directly (not
through the library), and have it run with the prescaler set to 1.
Ideally, you want the period of the timer in clock cycles to be:
F_CPU / 15 kHz = 16,000 kHz / 15 kHz ≈ 1066.67 CPU cycles
If you round this to the nearest integer, you get
F_CPU / 1,067 = 16,000 kHz / 1,067 ≈ 14.9953 kHz
This is about 0.03% too slow, well within the tolerance of the ceramic
resonator clocking the Uno.
Here is my attempt at this. I have tested it.
constexpr float PWM_FREQUENCY = 15e3;  // 15 kHz
constexpr uint16_t PERIOD = round(F_CPU / PWM_FREQUENCY);

void setup() {
    // Configure Timer 1 for 15 kHz PWM on pin 9 = PB1 = OC1A.
    DDRB  |= _BV(PB1);    // set pin as output
    TCCR1B = 0;           // stop timer
    TCCR1A = _BV(COM1A1)  // non-inverting PWN on OC1A
           | _BV(WGM11);  // mode 14: fast PWM, TOP = ICR1
    TCCR1B = _BV(WGM12)   // ditto
           | _BV(WGM13)   // ditto
           | _BV(CS10);   // clock @ F_CPU
    ICR1   = PERIOD - 1;  // period
    OCR1A  = PERIOD / 4;  // duty cycle
}

void loop(){}