This paper contains a complete solution of a problem of great importance for the theory of Dirichlet series, which was formulated by \textit{H. Bohr} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1913, 441--488 (1913; JFM 44.0306.01)] but during 18 years resisted the efforts of several mathematicians. The main result of the paper is that for any \( \sigma \) satisfying the condition \( 0 \le \sigma \le \frac{1}{2} \) there exists an ordinary Dirichlet series \(\sum_{n=1}^{\infty} a_n n^{-s} \) for which the width of the strip of uniform but not absolute convergence, \(d=\sigma_a - \sigma_u\) is exactly equal to \(\sigma\). It was proved by Bohr that \( d \leqq \frac{1}{2} \) and by \textit{O. Toeplitz} [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1913, 417--432 (1913; JFM 44.0405.02)] that \( \max d \ge \frac{1}{4} \). The authors base their proof upon an ingenious combination and extension of various tools developed by Bohr: the relationship between the Dirichlet series \[ \sum_{n=1}^{\infty} a_n n^{-s} \] and the associated power series in infinitely many variables \[ P\left(x_1, y_2, \ldots\right)=\sum_{n=1}^{\infty} a_n x_{n_1}^{\nu_1} \ldots x_{n_r}^{\nu_r},\quad n=p_{n_1}^{\nu_1} \ldots p_{n_r}^{\nu_r} \] where \( p_1, p_2, \ldots, \) is the sequence of consecutive primes \( 2,3,5, \ldots ; \) by Toeplitz: usage of matrices \( \left(a_{s t}\right) \) satisfying the conditions \( \sum_{t=1}^{n} a_{r t} \bar{a}_{s t}=n \delta_{r s} \) and of other matrices that are obtained from the preceding one by ``substituting'' a matrix \( \left(b_{p q}\right) \) of the same type, \( \left(a_{s, t}\left(b_{p q}\right)\right) ; \) by Littlewood: bilinear forms \( \sum_{s_{1}}^{\infty} a_{s t} x_{s}^{(1)} x_{t}^{(2)} \) bounded in the space \( \left(G_{0}\right)\left|x_{1}^{(i)}\right| \le 1,\left|x_{2}^{(i)}\right| \le 1, \ldots . \) In \( \S 1--3 \) the authors extend the results of Littlewood \( (m=2) \) to \(m\)-linear, symmetric \(m\)-linear and \(m\)-ic forms. After defining an \(m\)-linear form \[ L\left(x^{(1)}, \ldots, x^{(m)}\right) \equiv \sum_{i_{1}} \equiv \sum_{i_{2}}^{\infty} a_{i_{1} i_{2}}, \ldots, i_{m} x_{i_{1}}^{(1)} x_{i_{2}}^{(2)} \ldots x_{i_{m}}^{(m)} \] to be bounded by \( H \) in \( \left(G_{0}\right) \) when all the segments \[ \sum_{i_{1}=1}^{N_{1}} \cdots \sum_{i_{m}=1}^{N_{m}} a_{i_{1}} \ldots a_{i_{m}} x_{i_{1}}^{(1)} \ldots x_{i_{m}}^{(m)} \] remain absolutely \( \leqq H \) in \( \left(G_{0}\right), \) the authors prove that a necessary condition that \( L\left(x^{(1)}, \ldots, x^{(m)}\right) \) be bounded in \( \left(G_{0}\right) \) by \( H \) is that the quantities \( S \) and \( T^{(1)}, \ldots, T^{(m)} \) \[ S=\left[\sum_{i_{1}, \ldots, i_{m}=1}^{\infty}\left|a_{i_{1}, \ldots, i_{m}}\right|^{\varrho}\right]^{\frac{1}{\varrho}} ; \quad T^{(\nu)}=\sum_{i_{\nu}=1}^{\infty} T_{i_{\nu}}^{(\nu)} ; \quad T_{i_{\nu}}^{(\nu)}=\left[\sum\left|a_{i_{1}, \ldots, i_{m}}\right|^{2}\right]^{\frac{1}{2}} ; \quad \varrho=\frac{2 m}{m+1} \] where the last summation is extended over \( i_{1}, i_{2}, \ldots, i_{\nu-1}, i_{\nu+1}, \ldots, i_{m} \) from 1 to \( \infty, \) be less than \( A_{m} H, A_{m} \) being a constant which depends only on \( m.\) It is also proved that the result above is the ``best'' possible, the same being true for the symmetric \( m \)-linear and \( m \)-ic forms. An important rôle in the proof is played by the forms of the type \[ \sum_{i_1, \ldots, i_m=1}^n a_{i_1 i_2} \cdots a_{i_{m-1} i_m} x_{i_1}^{(1)} \cdots x_{i_m}^{(m)} \] where the matrix \( \left(a_{rs}\right) \) is obtained from a simpler matrix \( \left(M_1 =\exp 2 \pi irs / p\right) \) \( r, s=1,2, \ldots, p\), \(p\) a prime \( >m, \) by successive substitutions \[ \left(a_{r s}\right)=\left(M_{\mu}\right)\left(r, s=1,2, \ldots, p^{\mu}\right), \quad\left(M_{\mu}\right)=\left(\exp (2 \pi i r s / p)\left(M_{\mu-1}\right)\right) \] In \( \S 4 \) the authors prove an important extension of a classical result of Bohr: If the power series \[ P\left(x_{1}, x_{2}, \ldots\right)=c+\sum_{i=1}^{\infty} c_{i} x_{i}+\cdots+\sum_{i_1, \ldots=1}^\infty c_{i_1,\ldots, i_m} x_{i_1}\cdots x_{i_m}+ \cdots \] is bounded in the domain \( \left|x_{n}\right| \leqq G_{n}=p_{n}^{-\sigma_{0}}, n=1,2, \ldots, \) then its \( m \) -th polynomial \[ P_{m}\left(x_{1}, x_{2}, \ldots\right)=c+\sum c_{i} x_{i}+\cdots+\sum c_{i_{1}}, \ldots, i_{m} x_{i_{1}} \ldots x_{i_{m}} \] is absolutely convergent in \( \left|x_{n}\right| \leqq \varepsilon_n G_n, \) provided the series \( \sum \varepsilon_{\varepsilon_{n}}^{\sigma_{m}}, \sigma_{m}=2 m / m-1 \) converges. It is proved next that this result is the best possible. Thus it is shown that examples of Toeplitz are the best obtainable in the case where the associated power series \( P \) reduces to a quadratic form \( (m=2) . \) In \( \S 5-6 \) the main result of the paper is derived. The derivation is made comparatively easy by the preceding preparations. In the concluding \( \S 7 \) an extension to some more general Dirichlet series is indicated.