Force download a pdf link using javascript/ajax/jquery

Use the HTML5 "download" attribute

<a href="iphone_user_guide.pdf" download="iPhone User's Guide.PDF">click me</a>

2021 Update: Now supported in all major browsers! see: caniuse.com/#search=download

If you need to support older browsers, or you're looking for an actual javascript solution (for some reason) please see lajarre's answer

Here is a Javascript solution (for folks like me who were looking for an answer to the title):

function SaveToDisk(fileURL, fileName) {
    // for non-IE
    if (!window.ActiveXObject) {
        var save = document.createElement('a');
        save.href = fileURL;
        save.target = '_blank';
        save.download = fileName || 'unknown';

        var evt = new MouseEvent('click', {
            'view': window,
            'bubbles': true,
            'cancelable': false
        });
        save.dispatchEvent(evt);

        (window.URL || window.webkitURL).revokeObjectURL(save.href);
    }

    // for IE < 11
    else if ( !! window.ActiveXObject && document.execCommand)     {
        var _window = window.open(fileURL, '_blank');
        _window.document.close();
        _window.document.execCommand('SaveAs', true, fileName || fileURL)
        _window.close();
    }
}

source: http://muaz-khan.blogspot.fr/2012/10/save-files-on-disk-using-javascript-or.html

Unfortunately this is not working for me with IE11, which is not accepting new MouseEvent. I use the following in that case:

//...
try {
    var evt = new MouseEvent(...);
} catch (e) {
    window.open(fileURL, fileName);
}
//...