Would suggest not to mix jQuery and pure JavaScript. All you need is to add the keyup keydown instead of input:

$('table input').on('keyup keydown', function() {
  var parent = $(this).closest('tr');
  var value = parent.find('[name="item_value[]"]').val();
  var amount = parent.find('[name="quantity[]"]').val();
  var total_cost = value * amount;
  parent.find('[name="total_cost[]"]').val(total_cost);
  total_value = total_value + total_cost;
  $('#total').val(total_value);
});

Your code works for me (see below). Maybe check where you are binding your keyup event. It should be bound once when the document loads before the page shows. If you bind it multiple times (i.e. if the code that contains your keyup function runs more than once) you will run into problems.

$("#arama").on("keyup", function(event) {
  var i = event.keyCode;
  if ((i >= 48 && i <= 57) || (i >= 96 && i <= 105)) {
    $("#arama").off("keyup");
    console.log("Number pressed. Stopping...");
  } else {
    console.log("Non-number pressed.");
  }
});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>

<input id="arama" />

Also the url in the ajax call does not support ~/. You need to use razor syntax to generate, or the best sotionis to set the base href in html

apply the class of all of your expenses input like

 <input type="text" class="expenses" />

then get the total value like this.

$('.expenses').on('keyup', function () {
   let totalExpense = 0;
    $(document).find("input.expenses").each(function(){
       totalExpense += parseFloat($(this).val())
   })
});

As Math already stated, you can create a function that runs as per key press (or release in your case). The function should refine the search. I would recommend JS for this, but if you want to use jQuery, then here is what you could do:

$("#mySearch").on("keyup", function()
{
    // Refine search here
});

Or alternatively, you could just use the .keyup() function:

$("#mySearch").keyup(function()
{
    // Refine search here
});

Hope this helps