On a problem of Ramachandra and approximation of functions by Dirichlet polynomials with bounded coefficients

Theorem 1. Suppose $\Phi(n)$ is an increasing positive function and H > 0. Then $$\lim_{N \to \infty} \min_{|a_n| \le \Phi(n)} \int_0^H \left| 1 + \sum_{n=2}^N a_n n^{it-1} \right|^2 dt > 0,$$ if and only if $$\sum_{n=2}^{\infty} \frac{\log \Phi(n)}{n \log^2 n} < \infty.$$ Like Ramachandra's original conjecture, this result will have applications on the Riemann zeta-function, although somewhat weaker ones. For example in a forthcoming paper [5] we prove that $$\inf_{T} \int_{T}^{T+\delta} |\zeta(1+it)| dt = \frac{\pi^2 e^{-\gamma}}{24} \delta^2 + O(\delta^4), \qquad (\delta > 0), \tag{2}$$ We remark that this gives a positive lower bound that is independent of T. This implies that the Riemann zeta-function is not universal on the line $\mathrm{Re}(s)=1$ , since if the Riemann zeta-function was universal on $\mathrm{Re}(s)=1$ it should be possibly to approximate an arbitrarily small function by the Riemann zeta-function on that line. It is clear that Theorem 1 will also give us a lower bound independent of T in (2), by choosing $\Phi(n)=1$ and approximating the Riemann zeta-function by a truncated Dirichlet polynomial. While the constant will not be explicitly given in $\delta$ , and thus not give as sharp result as (2), Theorem 1 do allow us to prove corresponding lower bounds, and non universality on curves $\gamma(t)=\omega(t)+it$ whenever $\omega(t)$ tends to 1 sufficiently fast when $t\to\infty$ . We will also generalize Theorem 1 to more general Dirichlet polynomials. For example the sum over integers n can be replaced with sum over primes p or over shifted integers $n + \alpha$ . Also we may allow coefficients that can be quite general. Simple interesting cases includes divisor functions and Fourier coefficients of Maass wave forms. We will investigate this more carefully later in the paper. For now we just remark that a version of Theorem 1 for the shifted integers $n + \alpha$ implies the following result: $$\int_{T}^{T+\delta} |\zeta(1+it,\alpha)| dt \ge C_{\delta} > 0, \tag{3}$$ from which the fact that the Hurwitz zeta-function is not universal on the line Re(s) = 1 follows. The method we use in [5] for the Riemann zeta-function requires some multiplicative property, such as the function has an Euler product. The method from this paper does not require any such result however, although the proof in this paper will be somewhat more indirect and not give explicit estimates of $C_{\delta}$ such as Eq. (2). For an effective version of the method used in this paper and some explicit estimates of $C_{\delta}$ in (3), see our forthcoming paper [3].

Theorem 2. Suppose $0 = \lambda_0 < \lambda_1 < \cdots$ satisfy the Dirichlet condition. Let $A_0 = 1$ and $A_n$ , $n \geq 2$ be positive real numbers. Define $$\Lambda(x) = \sum_{\lambda_n \le x} A_n,$$ and suppose that $$\int_{1}^{\infty} \frac{\log \Lambda(x)}{x^2} dx < \infty.$$ Then we have for each H > 0 that $$\lim_{N \to \infty} \min_{|a_n| \le A_n} \int_0^H \left| 1 + \sum_{n=1}^N a_n e^{-i\lambda_n t} \right| dt > 0.$$

Lemma 1. Let $\lambda_n$ fulfill the Dirichlet condition (5), and let $B_n$ be a sequence of positive numbers such that $$\sum_{n=1}^{\infty} B_n < \infty.$$ Then for any H > 0 we have $$\inf_{|b_n| \le B_n} \int_0^H \left| 1 + \sum_{n=1}^\infty b_n e^{-\lambda_n it} \right| dt = \delta > 0.$$

Lemma 2. (Paley-Wiener) Suppose S(x) is a positive increasing function such that $$\int_0^\infty \frac{\log S(x)dx}{1+x^2} < \infty.$$ Then given any $\epsilon > 0$ there exists an entire function $\phi(x)$ of finite type $\epsilon$ such that $$\phi(x) \le \frac{1}{S(|x|)}, \qquad x \in \mathbb{R}.$$

Lemma 3. (Paley-Wiener) Suppose $\phi(x)$ is an entire function of exponential type A such that $$\int_{-\infty}^{\infty} |\phi(x)|^2 dx < \infty.$$ Then the Fourier-transform $\hat{\phi}$ will have support on [-A, A]. A direct consequence of Lemma 2 and Lemma 3 is the following:

Lemma 4. Let $\epsilon > 0$ and suppose S(x) is a positive increasing function such that $$\int_0^\infty \frac{\log S(x)dx}{1+x^2} < \infty.$$ Then there exists a continuous function f with support on $[0, \epsilon]$ such that $\hat{f}(0) \neq 0$ , and such that $$|\hat{f}(t)| \le \frac{1}{S(|t|)}, \quad t \in \mathbb{R}.$$

Theorem 3. Let $A_n$ and $\Lambda(n)$ be defined as in Theorem 2. Suppose $$\int_{1}^{\infty} \frac{\varepsilon(x)}{x} dx < \infty,$$ for some positive decreasing function $\varepsilon(x)$ and that $$\Lambda(X) \ll \Lambda(X+Y) - \Lambda(X), \qquad (\varepsilon(X) \ll Y \ll 1)$$ (11) for some $\delta > 1$ . Let H > 0. Then the set of Dirichlet polynomials $$\left\{ \sum_{n=2}^{N} a_n e^{-\lambda_n i t}, |a_n| \le A_n \right\}$$ is dense in $L^2(0,H)$ if and only if $$\int_{1}^{\infty} \frac{\log \Lambda(x)}{x^2} dx = \infty. \tag{12}$$ Furthermore the conclusion holds true if we have the additional assumption that $$\sum_{n=1}^{\infty} A_n^2 < \infty$$ is convergent and $|a_n| \leq A_n$ is replaced by $|a_n| = A_n$ . Also, under this assumption we may replace the set of Dirichlet polynomials with the set of convergent Dirichlet series in $L^2(0,H)$ such that $|a_n| = A_n$ . Remark 3. We can for example choose $\varepsilon(x) = x^{-\delta}$ or $\varepsilon(x) = \log(x+1)^{-1-\delta}$ for some $\delta > 0$ in Theorem 3, and these examples indeed seems sufficient for the applications we consider in this paper. In general it is an interesting problem to try to replace (11) with as weak condition as possible. Can we find some $\varepsilon(x)$ such that the integral is divergent but we still have Theorem 3?

Lemma 5. Let f(x) be an entire function of exponential type. Then $$\int_0^\infty \frac{\log^+|f(x)|dx}{1+x^2} < \infty \implies \int_0^\infty \frac{\log^-|f(x)|}{1+x^2} dx < \infty.$$

Lemma 6. Suppose f(x) is a a continuous function with compact support and that $\hat{f}(0) = 1$ . If $\inf_{\hat{f}(z)=0} |x-z| > \delta > 2\varepsilon(x)$ , then $$\min_{t \in [x, x + \varepsilon(x)]} \left| \hat{f}(t) \right| = \log \left| \hat{f}(x) \right| + O(\delta^{-1} \varepsilon(x) x). \tag{14}$$

Theorem 4. Suppose $A_n > 0$ and that $\lambda_n$ fulfill the Dirichlet condition (5). Let $$\Lambda(X) = \sum_{\lambda_n \le X} A_n, \quad and \quad \int_1^\infty \frac{\varepsilon(x)}{x} dx < \infty$$ for some positive decreasing function $\varepsilon(x)$ and suppose that $$\Lambda(X+Y) - \Lambda(X) \gg Y\Lambda(X),$$ $(\varepsilon(X) \ll Y \ll 1),$ for $X \geq X_0$ . Then we have that for any H > 0 $$\lim_{N \to \infty} \min_{|a_n| \le A_n} \int_0^H \left| 1 + \sum_{n=1}^N a_n e^{-\lambda_n it} \right|^2 dt > 0,$$ if and only if $$\int_{1}^{\infty} \frac{\log \Lambda(x)}{x^2} dx < \infty.$$ The same conclusion holds true if $\min_{|a_n| \leq A_n}$ is replaced by $\min_{|a_n| = A_n}$ under the additional assumption that $$\sum_{n=1}^{\infty} A_n^2 < \infty.$$

Theorem 5. Suppose that $A_n$ are positive numbers such that $$\frac{1}{M} \sum_{n=T}^{T+M} A_n \approx \Phi(T), \qquad T/(\log \log T)^{1+\delta} \le M \le T.$$ for some $\delta > 0$ , $T \geq T_0$ and some positive increasing function $\Phi(n)$ . Then $$\lim_{N \to \infty} \min_{|a_n| \le A_n} \int_0^H \left| 1 + \sum_{n=2}^N a_n n^{it-1} \right|^2 dt > 0,$$ if and only if $$\sum_{n=2}^{\infty} \frac{\log \Phi(n)}{n \log^2 n} < \infty.$$

Corollary 1. Suppose $\Phi(p)$ is an increasing positive function and H>0. Then $$\lim_{N \to \infty} \min_{|a_p| \le \Phi(p)} \int_0^H \left| 1 + \sum_{\substack{p \text{ prime.} \\ p \le N}} a_p p^{it-1} \right|^2 dt > 0,$$ if and only if $$\sum_{p \ prime} \frac{\log \Phi(p)}{p \log p} < \infty.$$ We also mention the case of divisor coefficients:

Corollary 2. Suppose $\Phi(n)$ is an increasing positive function and H>0. Then $$\lim_{N \to \infty} \min_{|a_n| \le \Phi(n)} \int_0^H \left| 1 + \sum_{n=2}^N a_n d(n) n^{it-1} \right|^2 dt > 0$$ if and only if $$\sum_{n=2}^{\infty} \frac{d(n)\log\Phi(n)}{n(\log n)^3} < \infty.$$ In fact most known cases of arithmetic functions can be used as coefficients, divisor coefficients $d_k(n)$ , Fourier coefficients of cusp-forms and so forth.

Theorem 6. Suppose $\alpha > 0$ , and that $\Phi(n)$ is an increasing positive function and H > 0. Then $$\lim_{N \to \infty} \min_{|a_n| \le \Phi(n)} \int_0^H \left| \alpha^{it-1} + \sum_{n=1}^N a_n (n+\alpha)^{it-1} \right|^2 dt > 0,$$ if and only if $$\sum_{n=1}^{\infty} \frac{\log \Phi(n)}{n \log^2 n} < \infty.$$

Theorem 7. Suppose $\omega(t) \leq 1$ is an increasing function such that $$\int_{2}^{\infty} \frac{1 - \omega(t)}{t \log t} dt < \infty.$$ Then for each $\delta > 0$ there exists a $C_{\delta} > 0$ such that $$\int_{T}^{T+\delta} |\zeta(\sigma+it)| dt \ge C_{\delta}, \qquad \omega(T) \le \sigma.$$

Theorem 8. Suppose $\alpha > 0$ and that $\omega(t) \leq 1$ is an increasing function such that $$\int_{2}^{\infty} \frac{1 - \omega(t)}{t \log t} dt < \infty.$$ Then for each $\delta > 0$ there exists a $C_{\delta} > 0$ such that $$\int_{T}^{T+\delta} |\zeta(\sigma+it,\alpha)| dt \ge C_{\delta}, \qquad \omega(T) \le \sigma.$$

Corollary 3. Let α > 0. Then $$\int_{T}^{T+\delta} |\zeta(1+it,\alpha)| dt \ge C_{\delta} > 0,$$ This is the first result that shows a a lower bound in this problem for the Hurwitz zeta-function that is independent of T, and thus that the Hurwitz zeta-function is not universal on the line Re(s) = 1. For the special case of the Riemann zeta-function and for an argument for why this implies non universality and what universality on a line means, see the discussion in [\[5,](#page-19-4) pp. 5-6].