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Consider the function $$F(z) := \cos(z) \cos(z/3) \cos(z/5) \cos(z/7) \cdots$$ Note that $\cos(z/n) = 1 - \frac{z^2}{n^2} + \cdots$ so the product is absolutely convergent to an entire function.

I would like to know if there has been some study of this function, in particular growth rates on the real axis and in general upper and lower bounds in magnitude on horizontal and vertical lines.

The motivation is that this function has the same asymptotic number of zeros as the completed zeta function, $\xi(1/2+ i z)$, but the zeros are much more widely spaced. In fact the zeros are at $\frac{\pi}2 m$, $m$ odd, with degree being the number of odd divisors. Therefore I could guess the average behavior to be like that of $\xi(1/2+iz)$ (which is tightly bounded for $|\Im(z)|>1$), with the ratio of these two functions converging to 1 on vertical lines, but being unbounded above and below on all horizontal lines.

If not for this product, is there some similar product of cosines that leads to a nice function? Like how there is the telescoping product $$\prod_{n=1}^{\infty} \cos(z/2^n) = \sin(z) / z$$

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The infinite product $$\varrho(x) = \prod_{n = 1}^\infty \cos(\pi x / n),$$ and one integral involving it, were studied in the papers:

[1] B. Schmuland, Random harmonic series, Amer. Math. Mon. 110 (5) (2003) 407–416. https://doi.org/10.2307/3647827

[2] S. Bettin, G. Molteni, C. Sanna, Small values of signed harmonic sums, C. R. Math. Acad. Sci. Paris 356 (11–12) (2018) 1062–1074. https://doi.org/10.1016/j.crma.2018.11.007

in relation to a kind of harmonic series with random signs.

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    $\begingroup$ I'd like to notice that $\varrho(x/2) = \prod_{n = 1}^\infty \cos(\pi x /(2 n))$, so $ \prod_{n = 0}^\infty \cos(\pi x /(2 n+1)) = \frac {\varrho(x)} {\varrho(x/2)}$. $\endgroup$
    – user64494
    Commented Jun 21, 2021 at 10:28
  • $\begingroup$ Thanks for the references! This paper from the bibliography has a great analysis of the growth rate in sectors: Offner, C. D. (1980). Zeros and growth of entire functions of order 1 and maximal type with an application to the random signs problem. Mathematische Zeitschrift, 175(3), 189–217. doi:10.1007/bf01163024 link.springer.com/article/10.1007/BF01163024 $\endgroup$
    – Ralph Furman
    Commented Jun 21, 2021 at 22:06
  • $\begingroup$ This leaves open the behavior on the real axis (on horizontal lines). [2] shows subexponential decay, but I'll have to read it through more carefully to see if this is expected to be tight, or if it may have exponential decay $\endgroup$
    – Ralph Furman
    Commented Jun 21, 2021 at 22:07

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