Dependencies

Let \(q\equiv 7\pmod8\) be prime and \(\chi =\chi _{q,s}\) with \(s=\pm 1\). Then \(S_\chi \bigl(\tfrac {q^j-1}{2}\bigr)=0\) for every \(j\ge 1\), and Conrey’s function \(f_q(x)=\sum _{n\ge 1}(\frac nq)\sin (2\pi nx)/n^2\) vanishes identically on the interval \(\bigl[\frac{q-1}{2q},\frac{q+1}{2q}\bigr]\).

Let \(R_q=\{ 0\le r{\lt}q:\ T_q+q\, C(r)=0=T(r)\} \). Every \(m\ge 1\) whose base-\(q\) digits all lie in \(R_q\) has \(S_\chi (m)=0\), for both values of \(s\).

1. If \(q\equiv 1\pmod4\), then \(0,q-1\in R_q\), and \(S_\chi \) has at least \(2^J-1\) zeros below \(q^J\), hence at least \(\tfrac 12X^{\log 2/\log q}-1\) below \(X\).

2. If \(q\equiv 7\pmod8\), then \((q-1)/2\in R_q\), and \(S_\chi \bigl(\tfrac {q^j-1}{2}\bigr)=0\) for every \(j\ge 1\).

3. If \(q\equiv 3\pmod4\), \(q{\gt}3\), \(h(-q)=1\) and \(s=1\), then \(S_\chi (z)=0\) along \(z_0=q\), \(z_{i+1}=q(z_i+1)\). For \(q=3\) and \(s=1\) the same holds with \(27\) in place of \(q\): \(z_0=27\), \(z_{i+1}=27(z_i+1)\).

4. For every \(q\) and \(s\): if \(S_\chi (qk)=0\) and \(C(r)=T(r)=0\), then \(S_\chi (qk+r)=0\). When \(q{\gt}3\) and \(\chi (2)=\chi (3)=-1\) this applies with \(r=4\), since \(1-2-3+4=0\). For \(q=3\) and \(s=-1\), where also \(\chi (2)=\chi (3)=-1\), the conclusion holds as well: \(S_\chi (3k)=0\) implies \(S_\chi (3k+4)=0\).

5. If \(s=-1\), then \(S_\chi (q^2m)=q^2S_\chi (m)\) for every \(m\): a single zero of \(S_\chi \) gives infinitely many.

6. If \(q\equiv 3\pmod8\), then \(S_\chi \bigl(\tfrac {q^j-1}{2}\bigr)=h(-q)\, q^j\, (1+s+\dots +s^{j-1})\) for every \(j\ge 1\), with \(h(-3)\) read as the weighted class number \(1/3\). In particular, for \(s=-1\), \(S_\chi \bigl(\tfrac {q^{2i}-1}{2}\bigr)=0\) for every \(i\ge 1\).

Let \(n=2m+1\). The row \(m\) is non-empty if and only if \(\bigl(\frac2n\bigr)=1\), and then it is infinite.

Let \(q\) be an odd prime, \(\chi \) an odd Dirichlet character modulo \(q\) of any order, extended completely multiplicatively by any root of unity \(\chi (q)\), and \(S_\chi (m)=\sum _{n\le m}n\chi (n)\). Then \(S_\chi \bigl(\tfrac {q-1}2\bigr)=0\) if and only if \(\chi (2)=1\), and in that case \(S_\chi \bigl(\tfrac {q^j-1}2\bigr)=0\) for every \(j\ge 1\). The characters for which this happens are the odd characters trivial on \(\langle 2\rangle \).

Write \(\Sigma (A)\) for the set of subset sums of a finite set \(A\) of positive integers.

1. If \(\Sigma (A)\supseteq [\alpha ,\beta ]\) and \(x\notin A\), \(0{\lt}x\le \beta -\alpha +1\), then \(\Sigma (A\cup \{ x\} )\supseteq [\alpha ,\beta +x]\). As \(\Sigma (A)\) is symmetric under \(x\mapsto P_A-x\), \(\Sigma (A)\supseteq [\alpha ,\beta ]\) with \(\beta \ge P_A/2\) implies \(\Sigma (A)\supseteq [\alpha ,P_A-\alpha ]\).

2. (pairs) Let \(q_0,u_1,v_1,\dots ,u_t,v_t\) be distinct positive odd integers, \(d_j=(v_j-u_j)/2\) with \(1\le d_j\le 1+d_1+\dots +d_{j-1}\), \(H=\sum d_j\), \(U=\sum u_j\), and \(q_0\le 2H+1\). Then \(\Sigma (\{ q_0,u_j,v_j\} )\supseteq [U+q_0-1,\ U+2H+1]\).

3. (Sárközy, with Lev’s constants [ BW21 , Thm. 4.23 ] ) If \(X\) is a set of \(n\) positive integers with maximum \(x^*\) and \(4x^*\le \ell \le n^2/(12\log _2(4x^*/n))\), then \(\Sigma (X)\) contains an arithmetic progression of \(\ell \) terms and step \(s\le 4x^*/n\), each term a sum of at most \(6\ell /n\) distinct elements of \(X\).

The subset-sum statement of Lemma 2.2(b), in the form of the Lean theorems complete_range and pairs_subset_sum: a sequence with \(d_j\le 1+\sum _{i{\lt}j}d_i\) represents every integer in \([0,\sum d_i]\) as a subset sum, and the pairs version used in the proof of Theorem A.

Give the primes outside \(Q\) independent uniform signs. Then

so some choice has \(|R_Q|\le B_*(m):=\tfrac 13m^{3/2}\sqrt{\log m+13}\).

Let \(R_q=\{ 0\le r{\lt}q:\ T_q+q\, C(r)=0=T(r)\} \). Every \(m\ge 1\) whose base-\(q\) digits all lie in \(R_q\) has \(S_\chi (m)=0\), for both values of \(s\).

Let \(q\equiv 3\pmod4\), \(q{\gt}3\), \(s=1\), \(h=h(-q)\) and \(D(r)=C(r)-h\). Suppose that \(B\) has base-\(q\) digits \(d_{L-1},\dots ,d_0\) with

Then, with \(Q=q^L\), every \(m_t=q\, B\, (1+Q+\dots +Q^{t-1})\), \(t\ge 1\), is a zero of \(S_\chi \).

Let \(q\equiv 3\pmod8\), \(q{\gt}3\), and let \(\chi _{q,-1}\) be the extension with value \(-1\) at \(q\). If \(\sum _{n\ge 1}\chi _{q,-1}(n)\sin (2\pi nx)/n^2\ge 0\) for \(0\le x\le 1/2\), then \(R_q=\varnothing \).

Let \(n=2m+1\) and let \(\varepsilon \) be any function on \(1,\dots ,m\) with values \(\pm 1\). Then

In particular, if \(\chi \) is a primitive quadratic character of conductor \(f\) with \(\gcd (f,m!)=1\), then \(T(m,\chi )\) does not vanish unless \(\bigl(\tfrac 2n\bigr)=1\).

Let \(F\) be a finite set of primes and \(\varepsilon _0:F\to \{ \pm 1\} \). There is \(M(F)\) such that for every admissible \(m\ge M(F)\) some \(\varepsilon \) with \(\varepsilon |_F=\varepsilon _0\) has \(S_\varepsilon (m)=0\). For \(F=\{ 2,3,5,7\} \) one may take \(M(F)=60000\).

For every \(m\ge 3\) with \(m\equiv 0\) or \(3\pmod4\) there is a completely multiplicative \(\varepsilon :\mathbb N\to \{ \pm 1\} \) with

Let \(f\) be odd, let \(\chi \) be an odd Dirichlet character modulo \(f\), and put \(B_{1,\chi }=f^{-1}\sum _{a=1}^{f}a\, \chi (a)\). Then

If moreover \(\chi \) is primitive, so that \(B_{1,\chi }\neq 0\), the sum vanishes if and only if \(\chi (2)=1\).

On \(\Omega =\{ \pm 1\} ^{\mathcal P}\), the sign patterns on the primes, let \(\mathcal Z_\infty \) be the set of \(\varepsilon \) with \(S_\varepsilon (m)=0\) for infinitely many \(m\). With the product topology, \(\mathcal Z_\infty \) is a dense \(G_\delta \). With independent fair signs, \(\Pr (\mathcal Z_\infty )=0\).

For \(k\ge 0\) and \(0\le r{\lt}q\),

Let \(q\in \{ 179,227\} \) and \(\chi =\chi _{q,1}\). Then \(S_\chi (m)\ne 0\) for every \(m\ge 1\). This answers the second part of the question on \(q\equiv 3\pmod8\) in the paper: there are primes \(q\equiv 3\pmod8\) for which \(\chi _{q,1}\) has no zero at all.