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Gerhard Paseman
Gerhard Paseman
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One of Mertens' theorems gives that

$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$

where $B$ is a constant near $0.26$ in value and $E(k)$ is an error term whose size is dominated by something close to $4/\log{k}$, when $k$ is large enough to make the sum meaningful.

I want to work with partial sums of the above with $j \lt p \leq k$, so that I can say the partial sum of the reciprocals of primes greater than $j$ and at most $k$ is $\log{(\log{k}/\log{j})} + E(j,k)$ where $j$ is not too small (perhaps $j \gt 3$ or $5$), $k$ not too large, say $j^\alpha \lt k \lt j^\beta$ where often $1 \lt \alpha \lt \beta \lt e$, and $E(j,k)$ is comfortably small. Unfortunately $4[1/\log{k} + 1/\log{j}]$ looks too big for me; I am hoping to have (for large enough $j$) $E(j,k)$ bounded by something that is $O(1/j)$ or better.

I have access to Mark Villarino's treatment of Mertens' theorem. As of 2005, it seems $E(k)$ is no better than $O(1/\log{k}^2)$ I also hope to obtain recent work of Pintz and Diamond on oscillations ia related formula which is Mertens product formula, but I do not see yet how it will me help me with this formula.

As I am still a tyro at number theory, I don't even know how realistic my hopes are for $E(j,k)$ to be $O(1/j)$. Can someone who is familiar with the recent literature give me a guide post? Either references or heuristics showing what sort of bounds to expect for $E(j,k)$ or even $E(k)$ would be welcome.

UPDATE 2012.07.24 I want to acknowledge the contributions of joro, Christian Elsholtz, and Eric Naslund. joro and Eric helped me realize that expecting $O(x^{-1/2})$ error even conditionally is expecting a bit much, and Christian helped me realize that Dusart still has some nice unconditional refinements. I will likely accept Christian's answer, but not before I do some computations of my own. In particular, Rosser and Schoenfeld have in Theorem 20 of their 1962 paper on functions relating to primes a nice difference of $2/(x^{1/2}\log x)$ which is valid for $1 \lt x \lt 10^8$ between lower and upper estimates for the sum of interest, and I may end up using or refining that estimate in combination with Dusart's results for larger $x$ for my own nefarious purposes. I am hoping in particular that the oscillations will be slow enough that my desired partial sums from $j$ to $j^\alpha$ will have very small error. END UPDATE 2012.07.24

Gerhard "Ask Me About System Design" Paseman, 2012.07.19

One of Mertens' theorems gives that

$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$

where $B$ is a constant near $0.26$ in value and $E(k)$ is an error term whose size is dominated by something close to $4/\log{k}$, when $k$ is large enough to make the sum meaningful.

I want to work with partial sums of the above with $j \lt p \leq k$, so that I can say the partial sum of the reciprocals of primes greater than $j$ and at most $k$ is $\log{(\log{k}/\log{j})} + E(j,k)$ where $j$ is not too small (perhaps $j \gt 3$ or $5$), $k$ not too large, say $j^\alpha \lt k \lt j^\beta$ where often $1 \lt \alpha \lt \beta \lt e$, and $E(j,k)$ is comfortably small. Unfortunately $4[1/\log{k} + 1/\log{j}]$ looks too big for me; I am hoping to have (for large enough $j$) $E(j,k)$ bounded by something that is $O(1/j)$ or better.

I have access to Mark Villarino's treatment of Mertens' theorem. As of 2005, it seems $E(k)$ is no better than $O(1/\log{k}^2)$ I also hope to obtain recent work of Pintz and Diamond on oscillations ia related formula which is Mertens product formula, but I do not see yet how it will me help me with this formula.

As I am still a tyro at number theory, I don't even know how realistic my hopes are for $E(j,k)$ to be $O(1/j)$. Can someone who is familiar with the recent literature give me a guide post? Either references or heuristics showing what sort of bounds to expect for $E(j,k)$ or even $E(k)$ would be welcome.

Gerhard "Ask Me About System Design" Paseman, 2012.07.19

One of Mertens' theorems gives that

$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$

where $B$ is a constant near $0.26$ in value and $E(k)$ is an error term whose size is dominated by something close to $4/\log{k}$, when $k$ is large enough to make the sum meaningful.

I want to work with partial sums of the above with $j \lt p \leq k$, so that I can say the partial sum of the reciprocals of primes greater than $j$ and at most $k$ is $\log{(\log{k}/\log{j})} + E(j,k)$ where $j$ is not too small (perhaps $j \gt 3$ or $5$), $k$ not too large, say $j^\alpha \lt k \lt j^\beta$ where often $1 \lt \alpha \lt \beta \lt e$, and $E(j,k)$ is comfortably small. Unfortunately $4[1/\log{k} + 1/\log{j}]$ looks too big for me; I am hoping to have (for large enough $j$) $E(j,k)$ bounded by something that is $O(1/j)$ or better.

I have access to Mark Villarino's treatment of Mertens' theorem. As of 2005, it seems $E(k)$ is no better than $O(1/\log{k}^2)$ I also hope to obtain recent work of Pintz and Diamond on oscillations ia related formula which is Mertens product formula, but I do not see yet how it will me help me with this formula.

As I am still a tyro at number theory, I don't even know how realistic my hopes are for $E(j,k)$ to be $O(1/j)$. Can someone who is familiar with the recent literature give me a guide post? Either references or heuristics showing what sort of bounds to expect for $E(j,k)$ or even $E(k)$ would be welcome.

UPDATE 2012.07.24 I want to acknowledge the contributions of joro, Christian Elsholtz, and Eric Naslund. joro and Eric helped me realize that expecting $O(x^{-1/2})$ error even conditionally is expecting a bit much, and Christian helped me realize that Dusart still has some nice unconditional refinements. I will likely accept Christian's answer, but not before I do some computations of my own. In particular, Rosser and Schoenfeld have in Theorem 20 of their 1962 paper on functions relating to primes a nice difference of $2/(x^{1/2}\log x)$ which is valid for $1 \lt x \lt 10^8$ between lower and upper estimates for the sum of interest, and I may end up using or refining that estimate in combination with Dusart's results for larger $x$ for my own nefarious purposes. I am hoping in particular that the oscillations will be slow enough that my desired partial sums from $j$ to $j^\alpha$ will have very small error. END UPDATE 2012.07.24

Gerhard "Ask Me About System Design" Paseman, 2012.07.19

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Gerhard Paseman
Gerhard Paseman
  • 13k
  • 3
  • 32
  • 63

Better error bounds for partial sums of reciprocals of primes?

One of Mertens' theorems gives that

$$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k)$$

where $B$ is a constant near $0.26$ in value and $E(k)$ is an error term whose size is dominated by something close to $4/\log{k}$, when $k$ is large enough to make the sum meaningful.

I want to work with partial sums of the above with $j \lt p \leq k$, so that I can say the partial sum of the reciprocals of primes greater than $j$ and at most $k$ is $\log{(\log{k}/\log{j})} + E(j,k)$ where $j$ is not too small (perhaps $j \gt 3$ or $5$), $k$ not too large, say $j^\alpha \lt k \lt j^\beta$ where often $1 \lt \alpha \lt \beta \lt e$, and $E(j,k)$ is comfortably small. Unfortunately $4[1/\log{k} + 1/\log{j}]$ looks too big for me; I am hoping to have (for large enough $j$) $E(j,k)$ bounded by something that is $O(1/j)$ or better.

I have access to Mark Villarino's treatment of Mertens' theorem. As of 2005, it seems $E(k)$ is no better than $O(1/\log{k}^2)$ I also hope to obtain recent work of Pintz and Diamond on oscillations ia related formula which is Mertens product formula, but I do not see yet how it will me help me with this formula.

As I am still a tyro at number theory, I don't even know how realistic my hopes are for $E(j,k)$ to be $O(1/j)$. Can someone who is familiar with the recent literature give me a guide post? Either references or heuristics showing what sort of bounds to expect for $E(j,k)$ or even $E(k)$ would be welcome.

Gerhard "Ask Me About System Design" Paseman, 2012.07.19