Sending Email with PHP and AJAX

I am creating a sign up form. Instead of using $_POST I used jQuery to get the form data and I'd like to send an email to the customer. The signup page is working property but i am having trouble to send an email with the data.

Here is the button to send the data

<button type="button" class="next action-button" id="fourthnext" >Register</button>

if($input.attr("id") == "fourthnext"){
//some code
 $.ajax({
          type: 'POST',
          url : path_to_functions,              
          data: data1,
          success: function(data){
            if (data!='') {
              finalerror.html(data);
              finalerror.show();
              fourthstepload.hide();
              show_result();
              //Disable the button and test the user to go back #TODO
              throw new Error();
            }else{
              finalerror.html(data);
              finalerror.show();
              fourthstepload.hide();
            }
          }
      });

show_result() function is called for any call flow the customer chooses so I think i can include the mail function in it. i have a simple mail php file for testing but its not even called.

var path_to_mail = path.href+'mailtest.php';
function show_result(){
    console.log("Debug start");
    $.ajax({
        type: "POST",
        url : path_to_mail,
        success: function(){
            console.log('success');
        },
        error : function(){
            console.log('error');
        }
    });
//rest of the code works properly such as
      $(".final_company").html($("#companyname").val());
      $("#final_login").html($("#inputEmail4").val());
      //hide previous step
      $(".hidemewhendone").hide();
      //show Result step
      $(".final_fieldset").show();
     

I could include more or the full code if the information I provided is not sufficient enough. Thank you

mailtest.php is literally just

<?php mail($myemail, "test", "message", $headers);?>

source https://stackoverflow.com/questions/69366487/sending-email-with-php-and-ajax