A

#### AfterMath

**Conjecture. (Montgomery)**$$\psi(x+h)-\psi(x)=h+O_{\varepsilon}(h^{\frac{1}{2}}x^{\varepsilon})$$ for $2\le h\le x$.

**My question: Which heuristic arguments support this conjecture (excluding numerical verification)?**

The conjecture appears in several places:

(A) H. L. Montgomery, "Problems concerning prime numbers", Proceedings of symposia in pure mathematics, Vol. XXVIII, pp.307-310, Mathematical developments arising from Hilbert Problems (1976), AMS. Providence, Rhode island. [See p.309]

(B) D. A. Goldston, "On a result of Littlewood concerning prime numbers", Acta Arithmetica, Vol.40 (1982), pp. 263-271. [See p.269]

(C) H. L. Montgomery, R. C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press (2007) [Conjecture 13.4, p.422]

**Following the argument of Montgomery and Vaughan (p.422),**is something I manage up to a certain point, but I'm not sure how to "get there". Specifically, using the explicit formula for $\psi(x)$ on the form $$\psi(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\frac{\zeta'}{\zeta}(0)-\frac{1}{2}\log(1-x^{-2})+\frac{1}{2}\Lambda(x) \hspace{5mm}(x>1),$$ we can write $$\psi(x+h)-\psi(x)=h-\sum_{|\gamma|\le T}C(\rho)+\lim_{U\to \infty}\sum_{T<|\gamma|\le U}C(\rho)+ O\big(\hspace{-0.2mm}\log\hspace{0.4mm}\max(x,h)\big)$$ for $x,h\ge 2$, say, where $$C(\rho)=\frac{(x+h)^{\rho}-x^{\rho}}{\rho} \ll \min\Big(hx^{\beta-1}, \frac{x^{\beta}}{|\gamma|}\Big).$$

Now assume that the Riemann hypothesis is true, and write \begin{align*}C(\rho)=&\;\int_{x}^{x+h}t^{\rho-1}dt=\int_{x}^{x+h}x^{\rho-1}dt+\int_{x}^{x+h}t^{\rho-1}-x^{\rho-1}dt\\ =&\; hx^{\rho-1}+\int_{0}^{h}(x+t)^{\rho-1}-x^{\rho-1}dt\\ =&\; hx^{\rho-1}+\int_{0}^{h}(\rho-1)\int_{0}^{t}(x+z)^{\rho-2}dzdt\\[1mm] =&\; hx^{-\frac{1}{2}+\gamma i}+O\big(h^2x^{-\frac{3}{2}}|\gamma|\big). \hspace{30mm} (\dagger) \end{align*} Ignoring the error here for the moment and taking $T=x/h$, then $$\sum_{|\gamma|\le x/h}hx^{-\frac{1}{2}+\gamma i}=\frac{h}{\sqrt{x}}\sum_{|\gamma|\le x/h}\text{e}^{\gamma \log(x)i}.$$ Now, if we were to replace the $\gamma \log x$-s with independent and identically distributed uniform random variables $(Y_n)_{n=1}^{\infty}$ on the interval $[0,2\pi)$ , then, as this this post shows, we could conjecture that the above sum behaves like $$\frac{h}{\sqrt{x}}\sum_{n\ll N(x/H)}\text{e}^{Y_n i}\ll_{\varepsilon} \frac{h}{\sqrt{x}}N\big(\frac{x}{h}\big)^{1/2+\varepsilon}\ll_{\varepsilon} \frac{h}{\sqrt{x}}\Big(\frac{x}{h}\log\big(\frac{x}{h}\big)\Big)^{\frac{1}{2}+\varepsilon}=h^{\frac{1}{2}-\varepsilon}x^{\varepsilon}\big(\hspace{-0.2mm}\log \frac{x}{h}\big)^{\frac{1}{2}+\varepsilon},$$ as in Montgomery's conjecture.

This is the point where I get stuck. For indeed, there are two unresolved sizes here. The first is the contribution of the large zeros: $$\sum_{|\gamma|>x/h}C(\rho), \hspace{15mm} (\ddagger)$$ and the second is the contribution of the error term in $(\dagger)$. The contribution of the error in $(\dagger)$ may not be too hard to resolve, but I am quite unable to show that the sum $(\ddagger)$ is of order $O_{\varepsilon}(h^{\frac{1}{2}}x^{\varepsilon})$. For example, using the explicit formula for $\psi(x)$ on the (commonly stated) form $$\psi(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}+O\Big(\frac{x\log^2(xT)}{T}+\log x\Big) \hspace{4mm} (x,T\ge 2),$$ I obtain $$\sum_{|\gamma|>x/h}C(\rho)\ll \frac{(x+h)\log^2((x+h)\frac{x}{h})}{x/h}\ll h\log^2 x$$ if $2\le h\le x$, which is not $O_{\varepsilon}(h^{\frac{1}{x}}x^{\varepsilon})$ unless $h\ll x^{2\varepsilon}(\log x)^{-4}$ (and this does not permit $2\le h\le x$ if, say, $0<\varepsilon<1/2$). The problem may be that the explicit formula with error term used here is unconditional, and that a better formula assuming RH should be employed.

Montgomery and Vaughan say about this (on p.422), that "The contribution of zeros with $|\gamma|>x/h$ can be attenuated by employing a smoother weight, but no amount of smoothing will eliminate the smaller zeros." By smoothing, they here likely mean that the explicit formula for $\sum_{n\le x}\Lambda(n)=\sum_{n=1}^{\infty}\Lambda(n)1_{(x,x+h]}(n)$ should be replaced by an explicit formula for $$\sum_{n=1}^{\infty}\Lambda(n)w(n),$$ where $w(n)=w(n;x,h)$ is a `weight function'. This weight function should be such that it gives a useful explicit formula (meaning that the contribution of the $\rho$-s in the right hand side decays rapidly as a function of $|\gamma|$), but also approximates the indicator function $1_{(x,x+h]}(n)$ to such an extent that $\sum_{n}\Lambda(n)w(n)$ is close to $\psi(x+h)-\psi(x)$.

I have been playing around with such explicit formulas lately, including the formulas \begin{align*} \frac{1}{k!}\sum_{n\le x}\Lambda(n)(x-n)^{k}=&\; \frac{x^{k+1}}{(k+1)!}-\frac{x^{k}}{k!}\frac{\zeta'}{\zeta}(0)-\sum_{\rho}\frac{x^{\rho+k}}{\rho(\rho+1)\cdots (\rho+k)}+ \sum_{0\le j\le (k-1)/2}\frac{x^{k-2j-1}}{(2j+1)!(k-2j-1)!}\frac{\zeta'}{\zeta}(-2j-1)\\[1.5mm] +&\; (-1)^{k}\sum_{j>k/2}x^{k-2j}\frac{(2j-k-1)!}{(2j)!}+\sum_{0<j\le k/2} \frac{x^{k-2j}}{(2j)!(k-2j)!}\Big(\frac{1}{2}\frac{\zeta''}{\zeta'}(-2j)-\log x+\sum_{\substack{r=-2j\\ r\ne 0}}^{k-2j}r^{-1}\Big) \hspace{5mm} (x\ge 1, k\in \mathbb{N}^{+}), \\[2mm] \frac{1}{\Gamma(\xi+1)}\sum_{n<x}\Lambda(n)(x-n)^{\xi}=&\; \frac{x^{\xi+1}}{\Gamma(\xi+2)}-\sum_{\rho}\frac{x^{\rho+\xi}\Gamma(\rho)}{\Gamma(\rho+\xi+1)}-\frac{x^{\xi}}{\Gamma(\xi+1)}\frac{\zeta'}{\zeta}(0)+\sum_{j=0}^{\infty} \frac{x^{\xi-2j-1}}{\Gamma(2j+2)\Gamma(\xi-2j+1)}\cdot \frac{\zeta'}{\zeta}(-2j-1)\\ -&\;\sum_{j=1}^{\infty} \frac{x^{\xi-2j}}{\Gamma(2j+1)\Gamma(\xi-2j+1)}\Big(-C_{\text{Eul}}+\sum_{k=1}^{2j}\frac{1}{k}+\frac{1}{2}\frac{\zeta''}{\zeta'}(-2j)+\log(x)-\psi^{(0)}(\xi-2j+1)\Big) \hspace{5mm} (x\ge 1, \text{Re }\xi>0, \xi \not \in \mathbb{Z}), \\[2mm] \sum_{n\le x}\Lambda(n)\log(x/n)=&\;x-\sum_{\rho}\frac{x^{\rho}}{\rho}-(\log 2\pi)\log x-(\frac{\zeta'}{\zeta})'(0)-\frac{1}{4}\sum_{k=1}^{\infty}\frac{x^{-2k}}{k^2} \hspace{5mm} (x>1),\\[2mm] \sum_{n=1}^{\infty}\Lambda(n)\text{e}^{-n/z}=&\; z-\sum_{\rho}\Gamma(\rho)z^{\rho}-\text{e}^{-1/z}\log 2\pi -(-1+\cosh 1/z)\log z+\sum_{k=1}^{\infty}(-1)^{k}\frac{\zeta'}{\zeta}(k+1)\frac{z^{-k}}{k!}\\ -&\;\sum_{k=0}^{\infty}\frac{\Gamma'}{\Gamma}(2k+2)\frac{z^{-2k-1}}{(2k+1)!} \hspace{5mm} (\text{Re }z>0). \end{align*} However, I am not able to get the desired result. Indeed, I am not able to combine the weight functions provided in the formulas above, to get a weight function approximating $1_{(x,x+h]}(n)$ to a reasonable extent while giving a reasonable explicit formula. This may be because:

- I have not found any literature explaining what would constitute a good smoothing of $1_{(x,x+h]}(n)$ in this case (i.e. how does the smoothness and cut-off play a role)
- Taking a weighted sum of one of the explicit formulas above, one could likely approximate $1_{(x,x+h]}(n)$ by expressions of the form $\sum_{n=1}^{N}a_nw(n,x_n,h_n)$. Here I am not at all sure which function spaces I would try to do an approximation in (e.g. which norm).

These investigations are related to my Master's thesis on primes in short intervals, and I would highly appreciate if anyone could comment on how Montgomery's conjecture can be backed up. (And possibly also what the goal/strategy of the smoothing process should be, if you would be kind enough to explain).

Sincerely, R.