5

[Added numbers 11/13.]

Motivation (can skip).

When prime powers $p_n$ are used to calculate

$$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$

for (say) $N= 30,$ $x>5$, at sign changes $(+/ -)$ $y(x)$ seems to give approximately the imaginary parts $\gamma_r$ of a subset of zeros of $\zeta(s).$ The first few zero indices, with some ambiguity when zeros are tightly spaced, are $r = 1, 2, 3, 4, 6, 7, 10, 11, 13, 16, 18,...$

It's a lacklustre approximation and I think this has been explained in other questions on this site. When the zeros (+/-) of (1) are used to form a well-known approximation of

$$\sum x^{\rho}/\rho$$

which in pertinent part is

$$\sum_r \frac{\sin (\gamma_r \log x)}{\gamma_r}\hspace{5mm}(2), $$

the resulting graph has discontinuities at the prime powers.

If instead of primes (prime powers) we use non-prime powers $q_r$ in (1), we will also get a sequence of approximate "zeros" at sign changes (+/-). If we use them instead of $\gamma_r$ in the approximation(2), we might expect to see discontinuities at non-prime powers. This is apparently the case.

Using 100 such "zeros" $\alpha_r$ the graph of

$$\sum_r \frac{\sin (\alpha_r \log x)}{\alpha_r} $$

appears to show discontinuities at 1,6,10,12,14,15,... in the same way that (2) shows discontinuities at prime powers.

Question.

When non-prime powers $q_n$ are used to form $Y(x)=\sum_{n=1}^{N}\frac{\sin (x \log q_n)}{q_n},$ the zeros $\alpha_r$ at sign changes (+/-), inserted in $G(x)=\sum_r \frac{\sin (\alpha_r \log x)}{\alpha_r},$ give discontinuities at approximately non-prime powers.

What meaning, if any, can be attached to the zeros $\alpha_r$ of $Y(x)$? Are they a subset of the complex parts of the zeros of some cousin of $\zeta(s)$?

Omitting $\alpha_n< 5$, in G(x) I used: 5.9, 7.3, 8.9, 12.2, 15.6, 19.9, 22.8, 26.0, 29.4, 31.5, 33.9, 36.6, 40, 42.2, 44.2, 47.2, 48.6, 50.85, 54, 55.7, 58.5, 61.6, 64.35, 66.2, 68.4, 70.15, 72.4, 75, 78, 79.9, 82.2, 83.9, 85.8, 89.5, 92.0, 93.8, 96.6, 99.6, 103.1, 104.65, 103.1, 104.7, 106.2, 107.8, 110.3, 113, 114.2, 117.9, 120.8, 124.9, 127.2, 131.6, 135.3, 137.6, 138.8, 140.7, 142, 143, 1, 145.5, 149.1, 151.8, 153.5, 155.8, 159.7, 163.2, 166.2, 168, 170.5, 173, 175.9, 176.2, 180.7, 184.2, 187.6, 189.9, 191.4, 194.5, 198.6, 200.9, 203.3, 205.9, 208.5, 211.5, 215.1, 216.6, 218.8, 222.3, 225.7, 226.7, 229, 230.5, 232.8, 234, 236.1, 241.8, 244.3, 246.6, 249, 250.2.

The image using the values above in $G(x)$ shows discontinuities at 1,6,10,12,14,15,18 but quickly deteriorates.

Community
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daniel
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  • Might be of interest: http://math.stackexchange.com/q/269997 – draks ... Sep 10 '15 at 07:00
  • @draks: I looked at that question and the very nice answers there (and long ago upvoted). The r.h.s. of (1) in Ray M's answer is quite different from (1) above. – daniel Sep 10 '15 at 08:27
  • but Ray's (1) comes close to the (2) from above... – draks ... Sep 10 '15 at 09:14
  • Ingham comments in Distr. of Prime Numbers that "the explicit formula [as set out in the comment below] suggests connections between the...discontinuities of $\psi(x)$ and the numbers $\rho.$ But no relationship essentially more explicit than [that one] has ever been established..." p.80. – daniel Jan 23 '19 at 11:41

1 Answers1

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There are quite a few questions (see 2185872, 573999, 424530, for example) that relate to this general topic and the short answer to many of them is that properties such as that asserted for the sum (2) in the question, if otherwise true, probably depend on the Riemann Hypothesis.

The earliest treatment I can find of sums zeros of $\zeta(s)$ and primes in the setting of transform-pairs is Guinand's A Summation Formula in the Theory of Prime Numbers, Proc. of the London Math. Society,Issue 1, Jan. 1948, pp. 107-119. He assumes RH throughout.

Ingham uses the sum $\sum_k \frac{\sin \gamma_k \log x}{\gamma_k}$ in his proof of $\Omega$ relations for $\pi(x)$ and $li(x)$ (etc.) without mentioning the behavior of the sum at primes.

daniel
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  • I would say the Riemann explicit formula $\sum_{p^k \le x} \log p = x-\sum_\rho \frac{x^\rho}{\rho} - C$ can be generalized to $\sum_{p^k \le x} p^{-sk}\log p = \frac{x^{1-s}}{1-s}-\sum_\rho \frac{x^{\rho-s}}{\rho-s} - B$. – user798409 Jan 10 '18 at 21:20
  • @user798409: thanks, yes for sure this is the source of all these. i wanted to mention the cite and credit ingham with the expression but a proper (long) answer would begin with your point. – daniel Jan 11 '18 at 07:10