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An alternative to using a StringBuilder is using LINQ with string.Join(). It's about as fast and is more concise and readable.

public string BinarySearchSetup(long[] data, long[] searchTerms)
{
    var resultString = "";

    var lowerBound = 0;
    var upperBound = data.Length - 1;
    var results = searchTerms.Select(x => BinarySearch(data, lowerBound, upperBound, x));
    return string.Join(" ", results);
}

If You want to avoid searching for duplicates, instead of working with a dictionary, you can just call Distinct(). If you want to parallelize the operation, you can just call AsParallel() and then AsOrdered() to preserve order:

var results = searchTerms
    .Distinct()
    .AsParallel()
    .AsOrdered()
    .Select(x => BinarySearch(data, lowerBound, upperBound, x));

If you want to cache the results, you can build a dictionary on the distinct elements in a similar fashion to the above LINQ query. AsOrdered() is not required because it's getting stored in a dictionary. The method will look like this:

public string BinarySearchSetup(long[] data, long[] searchTerms)
{
    var lowerBound = 0;
    var upperBound = data.Length - 1;
    var cache = searchTerms
        .Distinct()
        .AsParallel()
        .ToDictionary(x => x, x => BinarySearch(data, lowerBound, upperBound, x));
    var results = searchTerms.Select(x => cache[x]);
    return string.Join(" ", results);
}