The serial number 3,413,016 puts it into Block 1 3,407,088 to 3,437,087 of M1903A4 production. (ref. page 80 Collecting the American Sniper Rifle 1900 to 1945 by Joe Poyer (North Cape Publications Inc.) Yours has the "scant" stock by the way.
Only a few M1903A4 rifles had the "A3" marking changed to "A4", and apparently it was not supposed to happen. Some over-enthusiastic armorer figuring that he knew better than the U.S. Government (OK, he did but that is beside the point.) If an M1903A4 not longer was wanted as a sniper rifle, it was intended that by simply removing the scope and mount and fitting front and rear sights, that it became an M1903A3 and could then be issued as such.
I collect sniper rifles, mainly Canadian and British No. 4 (T), but have had several original M1903A4 (one factory original and the others arsenal overhauled and refinished.) I kept one M1903A4 with an original M73B1 scope for comparison and actually enjoy shooting it. I am looking for a replacement clicker spring for the drums by the way as my spring is broken at one end.
My advice:
1. Have the front sight removed e.g. by a gun smith. It should NOT be there for an M1903A4 as there is no rear sight on an A4.
2. I do suggest trying to buy an ORIGINAL correct scope and rings. The reference book I mentioned is very handy. It would have had an M73B1 in WWII service. An M73 (very rare), M81, M82 or M84 scope could also be appropriate e.g. as used in Korea, Vietnam etc. but the M73B1 is the CLASSIC. The left-handed sniper (Canadian actor) in "Saving Private Ryan" made the M1903A4 rifles very popular though they cheated and fitted two different scopes to it during filming! I always like to get a correct scope for a sniper rifle if it did not come with a correct one. Today you can find original scopes, but it will only become much harder and much more expensive over time. An M1903A4 with proper scope with have much greater collector value that a rifle with a civilian scope.
3. There are replica scopes being made, I but they lose value over time instead of increasing in value.
4. There are original Weaver 330 and M8 scopes, but again the proven M73B1 scopes is "always" right. Weaver 330 scopes on an M1903A4 are always supicious. The U.S. Army used a FEW very early on, and added a serial number, but this is so easy to fake on a cheap civilian 330 that collectors shy away from these unless there is some really solid evidence of original military use.
*Weaver 330 - There are two basic models, but the markings are the same. The U.S. Army liked the one with the drums that click. There are also the cone-shaped type, adjustable by screwdriver that are "silent". Collectors call these the 330C and 330S. Oddly the cone shaped silent type are shown in the 1943 manual on the M1903A4 ... just to confuse future collectors I expect. ;-)
*Weaver M8 - The M8 is scarce and I had one, but there is only one known reference to their use on the M1903A4 and that was related by the late Pete Senich. He and I used to correspond, but sadly there has not been any further confirmation of the M8 scope use by the military as far as I know.
Think about 19 years from now when someone looks at your rifle. "Wow! An ORIGINAL 100 year old M1903A4 sniper rifle from World War II !! Oh, too bad it has the wrong scope. You just can't find an original M73B1 scope these days. Too bad you didn't get one back in 2024 when they were availabe and cheap at around $1,000 to $1,500 compared to day's prices in 2043,, That is IF you can even find someone willing to sell one. Having an original M73B1 scope on it would have quadrulpled the value of your rifle."
How do I know? I bought my first No. 4 Mk. I (T) sniper rifle complete, matching and mint, in the wooden transit chest about 1971 for $75 dollars. Today that same rifle would sell for 100 times that, US $7,500. By the way, the No. 32, C No. 32 and especially the C No. 67 scopes for the No. 4 (T) sniper rifles are MUCH more expesive than an M73B1!
Your rifle is a gem and deserves a correct scope. Think of the old saying of Rolls Royce about their quality. "The quality remains long after the price is forgotten." Iamgibne if you had a lovely 1964 Mustang car, all original except for the engine. Imagine if you could buy a correct 1964 peiod engine for it. Expensive yes, but it would make that car so special!