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[on:]

The avr-gcc stdio.h gives specific advice.
printf() needs a valid stdout descriptor,
which needs a generic  putc() function.

Code:

// ---- place globally,  above setup() ---

FILE serialStdout = {0} ;

static int uart_write(char c, FILE *stream)
{
   loop_until_bit_is_set(UCSR0A, UDRE0) ;
   UDR0 = c;
   return 0;
}//end of global stuff

// ---- place after Serial.begin() invocation  ----

   /* fill in the FILE structure  */
   fdev_setup_stream( &serialStdout, uart_write,
                            NULL,_FDEV_SETUP_WRITE);

   /* reassign stdout */
   stdout = &serialStdout ;

   /* ... and test the result */
   printf("stdout's address is %p\n", stdout) ;

bon appetit.