Sums of prime powers

You are given positive integers N, m, and k. Is there a way to check if
$$\sum_{\stackrel{p\le N}{p\text{ prime}}}p^k\equiv0\pmod m$$
faster than computing the (modular) sum?

For concreteness, you can assume $e^k<m<N.$

I don’t know of a way, but it’s not obvious to me that no method exists. Fast ways to prove or refute the equivalence would be of interest. You can assume that the particular instance of the problem is ‘hard’, that is, the modulus is not close enough to N so that Rosser-type bounds on the sum would rule it out.

With $k=0$ this is just asking if $m|\pi(N)$, so it is possible to compute the sum in time $O(N^{1/2+\varepsilon})$ using the Lagarias-Odlyzko method. (Or more practically, one of the combinatorial $\pi(x)$ methods.) For $k>0$ the sum is superlinear and so cannot be stored directly (without, e.g., modular reduction) but it’s not clear whether a fast algorithm exists.

You can think of the problem as “Your friend, who has access to great computational resources, makes the claim (N, m, k). If her claim is true, can you prove it? If her claim is false, can you refute it?”.

Edit: I posted a related problem on cstheory, asking if there is a short proof or interactive proof that the sum is correct.

Solutions Collecting From Web of "Sums of prime powers"

Deléglise-Dusart-Roblot [1] give an algorithm which determines the number of primes up to $x$ that are congruent to $l$ modulo $k,$ in time $O(x^{2/3}/\log^2x).$ A modification of the algorithm of Lagarias-Odlyzko [2] allows the same to be computed in time $O(x^{1/2+o(1)}).$

So using either algorithm, find the number of primes in all residue classes mod primes up to $\log m.$ For each prime $q,$ take the total number of primes in each residue class times that residue class to the $k$-th power; this gives the value of
$$\sum_{\stackrel{p\le N}{p\text{ prime}}}p^k\pmod q.$$

Use the Chinese Remainder Theorem to determine the value of the sum mod $2\cdot3\cdots\log m.$ The Prime Number Theorem ensure that this, or a little more, is greater than $m$ and hence sufficient to determine the result uniquely. (Note that if $m>N^{k+1}/\log N$ or so, the calculations can be done exactly working mod $k\log N$ or so.)

This gives the sum (mod m or in Z) in time $O(N^{1/2+o(1)})$ since the number of calculations needed is logarithmic.

References

[1] Marc Deléglise, Pierre Dusart, and Xavier-François Roblot, Counting primes in residue classes, Mathematics of Computation 73:247 (2004), pp. 1565-1575. doi 10.1.1.100.779

[2] J. C. Lagarias and A. M. Odlyzko, Computing $\pi(x)$: An analytic method, Journal of Algorithms 8 (1987), pp. 173-191.

[3] Charles, answer on MathOverflow. (Yes, this is the same person. See the other answers there for different approaches.)